How much of the proton's mass is due to the Higgs field? The proton mass is 938 MeV. People often claim that

(A) The proton is a bound state of two up quarks and one down quark, with the three quarks contributing a total rest mass of $2 \times (2.2 \text{ MeV}) + (4.7 \text{ MeV}) = 9.1 \text{ MeV}$ (about 1% of the proton mass), and the other 99% coming from "QCD binding energy".

People further claim that

(B) The "QCD binding energy"'s contribution to the proton mass is independent of the Higgs field. Therefore, only 1% of the proton mass comes from the Higgs field.

Claim (A) is perhaps defensible as a heuristic explanation, although it sweeps a lot of subtleties under the rug. (For example, in a non-QCD context, binding energy (as usually defined) always decreases a bound state's energy below that of its constituent parts - the opposite of what happens in hadrons.) But as I understand it, claim (B) is simply incorrect.
As explained here, the proton ground state is best thought of not as a bound state of exactly three quarks, but instead as a superposition of many different huge collections of quarks and antiquarks, each of which has total up quark number 2 (i.e. two more up quarks than up antiquarks) and total down quark number 1 (i.e. one more down quark than down antiquark).
This is sometimes described as "two 'valence' up quarks and one 'valence' down quark, surrounded by a huge 'sea' of quark-antiquark pairs". In my opinion, this description is also misleading, because it implies that there are three specific "valence" quarks that are "real", while the other "sea" quarks are just "virtual" and physically distinct from the valence quarks. In fact, every individual quark is physically on the exact same footing. In particular, I believe that all of the quarks gain a mass contribution from the Higgs mechanism.
As Prof. Strassler explains in the link above, the proton mass is best throught of as arising from the sum of three contributing terms:


*

*The sum of the rest energies of all of the (many!) quarks and antiquarks

*The kinetic energy of the quarks, antiquarks, and gluons

*The binding energy stored in the gluon fields, which is negative and actually decreases the proton mass relative to what it would be if the quarks did not interact.


I believe that the Higgs mechanism is responsible for the contribution (1), which is much larger than the naive 9.1 MeV from just three quarks, and therefore much more than 1% of the proton mass. Is this correct?
Also, do we have any kind of quantitative estimate of the contribution of each of these three terms? The number of up quarks and the number of up antiquarks are individually indefinite, but can we numerically estimate their ground-state expectation values in order to estimate (1)? I know that QCD is strongly enough interacting that it's not particularly useful to think in terms of individual quarks, but I'm curious whether anyone's attempted that calculation.
 A: Yes, the standard model of a hadron as "three bound quarks" and therefore computing the contribution of the Higgs mechanism to the hadron mass solely from these three bound quarks is overly simplistic.
The name for the model of the proton that Strassler describes in the post you link is effectively the parton model, which was initially proposed without any knowledge of what the partons were, while we now identify them as quarks and gluons. The parton distribution functions describe the probability to detect a particular parton with a particular momentum at a particular overall energy scale.
And there's the first hindrance to resolve the question of "how much" the Higgs mechanism contributes to the mass: The parton distribution is not fixed, and depends on the overall energy scale (the energy scale is a measure for how "finely" we can "resolve" the substructure of the proton). There is no claim of a "true" parton distribution, it's all in flux and depends on the resolution power of the observer.
Secondly, we don't really know how to compute parton distribution functions, at least not when I last checked. We have to fit them to collider data - perturbative approaches are utterly worthless in this kind of strongly-bound QCD, and lattice calculations hit other roadblocks. I know that there are lattice models that in some situations predict cross sections from which you can extrapolate reasonable parton distributions, but I don't know that any such numerical computation is agreed upon to be "the" way to do it. A cursory search finds e.g. this paper about ab initio calculation of parton structure on the lattice, but someone with more expertise in the numerical aspects of the field will have to evaluate the merits of approaches like this.
Therefore, if we accept the parton model, I'd argue that we are forced to concede that the question of "how much" of the proton's mass is coming from where doesn't really make a lot of sense. In full QCD, an ab initio computation of hadronic masses (as in "Ab-initio Determination of Light Hadron Masses" by Durr et al.) is a bunch of really complicated lattice computation rather than something that would separate neatly into contributions to the mass of the three types you are looking for. Sure, the lattice model takes into account these three factors, but you don't get separate values for their contributions, the lattice model simply spits out a net hadron mass and that's it.
However, as the ab initio mass calculation I just linked shows, the sea quark contributions are highly relevant - the authors refer multiple times to how "light sea quark contributions" are necessary to make the result match the actual experimental masses. But I caution against trying to simply compute the difference between the lattice models with and without these contributions and assign that as "the sea quark contribution to mass", since the relation between lattice models and experiment is often much more subtle.
In "Proton Mass Decomposition from the QCD Energy Momentum Tensor" by Yang et al. it turns out that, while my pessimistic claims above are true in a naive sense, one can use lattice models that indeed do produce four distinct contributions to the hadron mass we naively expect. Note that the way in which they retrieve the actual contributions from the numerical results is rather convoluted and requires several intermediate quantities, passing through a parton distribution function in the form of the "momentum fractions" on the way. This lattice computation shows the following contributions:


*

*Quark condensate: 9%

*Quark energy: 32%

*Gluon energy: 36%

*Trace anomaly: 23%


We can note two things here: 
First, if all we're looking for is "how much of this is the Higgs", then the straightforward and false answer is that it is the quark condensate, because it's the only contribution where the masses of the quarks play a role. The gluon energy term includes the $\beta$-function for the QCD coupling, and as knzhou's answer rightly points out the (non-)existence of the Higgs massively affects the running coupling. So even after all this trouble we still haven't a "true" figure for how much of this is the Higgs, although we could certainly combine this with the 60% from knzhou's answer to get yet another value (see below for why that's not straightforward either).
Second, about one fourth of the mass comes from a place that the parton model doesn't even know about, namely the trace anomaly. The cause of this is the "instanton-filled vacuum" of QCD - the QCD vacuum is not a simple perturbative vacuum but instead a $\theta$-vacuum, which is the superposition of many instantonic vacua, and so fermions and the bosons of the gauge field carry a global anomaly term that contributes to the action, and hence the energy. 
The contribution of the Higgs to this part is murky because the paper does not compute the split contributions of the fermions and the gluons to this, but the $\beta$-function that depends on the Higgs only directly affects the gluonic anomaly. I'll leave to the reader to decide if this exercise actually brought us any closer to being able to talk about the Higgs' contribution to the hadron masses.
A: The present status is that  the proton mass comes "from four sources [2] known as the quark condensate ( ∼9%), the quark energy ( ∼32%), the gluonic field strength energy ( ∼37%), and the anomalous gluonic contribution ( ~23%)". https://physics.aps.org/articles/v11/118#
Only the first contribution of 9% depends on the quarks having mass. The Higgs contribution is minor, although larger than the naive figure based on the sum of the quark masses. 
A: I don't think it's valid to add up the rest masses of all the "sea" quarks and antiquarks -- a proton is a complex, strongly-coupled bound state of QCD, and I doubt one can just treat it as a bag of particles. The idea of a definite number of sea quarks probably doesn't make sense at all.
Here's a simpler argument that gets an intermediate estimate. Essentially by dimensional analysis, the mass of the proton is around $\Lambda_{\text{QCD}}$, which is around the GeV scale; we would expect to get a contribution of about this size whether the quarks were massive or not. 
The Higgs field affects $\Lambda_{\text{QCD}}$ because it affects the running of the strong coupling constant; recall that $\Lambda_{\text{QCD}}$ is defined as the point where this coupling becomes large. With the Higgs field on, heavy quarks "freeze out" when the RG scale passes their masses, causing the running to accelerate, since quarks provide a charge screening effect. With the Higgs field off, no quarks ever freeze out, so the running to be slower overall, and hence lowering $\Lambda_{\text{QCD}}$. In this post it's argued that this lowers $\Lambda_{\text{QCD}}$ by 40%, so the Higgs is responsible for 60% of baryonic mass.
A: 
How much of the proton's mass is due to the Higgs field?

My view on this is coloured by Gian Francesco Giudice’s 2010 book A Zeptospace Odyssey. On page 173 he said this: “The most inappropriate name ever given to the Higgs boson is ‘The God particle’. The name gives the impression that the Higgs boson is the central particle of the Standard Model, governing its structure. But this is very far from the truth”. He also said “the Higgs sector is rather arbitrary, and its form is not dictated by any deep fundamental principle. For this reason its structure looks frightfully ad hoc”. Giudice also said the mass of a body is the intrinsic energy of a body at rest, and tells us about E=mc². In his 1905 paper does the inertia of a body depend upon its energy-content?, Einstein said “the mass of a body is a measure of its energy-content”. No problem there.  
The problem is that the Higgs mechanism contradicts E=mc². Is the mass of a body a measure of its energy-content, or a measure of its interaction with some space-filling field? The idea for the latter comes from the electroweak sector of the standard model, where the weak force is said to be mediated by massive vector bosons. They have to be massive because the force is short range. See the Wikipedia Higgs mechanism article where you can read that “according to Goldstone’s theorem, these bosons should be massless”. If they aren’t, the bosons aren’t gauge bosons, and the theory isn’t a gauge theory. But here we are. We have Giudice saying “the Higgs mechanism accounts for about 1 per cent of the mass of ordinary matter, and for only 0.2 per cent of the mass of the universe. This is not nearly enough to justify the claim of explaining the origin of mass”. He also said the Higgs boson is “like the toilet of the Standard Model edifice”. As you are doubtless aware, Giudice is now head of theory at CERN. 

The proton mass is 938 MeV. People often claim that (A) The proton is a bound state of two up quarks and one down quark, with the three quarks contributing a total rest mass of $2 \times (2.2 \text{ MeV}) + (4.7 \text{ MeV}) = 9.1 \text{ MeV}$ (about 1% of the proton mass), and the other 99% coming from "QCD binding energy".

I see you've read Matt Strassler's article what's a proton anyway? The above claim is the original quark picture. That has morphed over the years. This picture from physorg hints at it:

Image from the PhysOrg article Physicists zoom in on gluons' contribution to proton spin

People further claim that (B) The "QCD binding energy"'s contribution to the proton mass is independent of the Higgs field. Therefore, only 1% of the proton mass comes from the Higgs field.

Yes, people like Gian Giudice. 

Claim (A) is perhaps defensible as a heuristic explanation, although it sweeps a lot of subtleties under the rug. (For example, in a non-QCD context, binding energy (as usually defined) always decreases a bound state's energy below that of its constituent parts - the opposite of what happens in hadrons.) But as I understand it, claim (B) is simply incorrect.

I think claim (A) is wrong, and I think claim (B) is wrong too. 

As explained here, the proton ground state is best thought of not as a bound state of exactly three quarks, but instead as a superposition of many different huge collections of quarks and antiquarks, each of which has total up quark number 2 (i.e. two more up quarks than up antiquarks) and total down quark number 1 (i.e. one more down quark than down antiquark).

Matt Strassler said the proton isn’t made up of three quarks joined together by three gluons. He says that’s a lie, a white lie, but a big one. Instead he said there’s “zillions of gluons, antiquarks, and quarks in a proton”, and gave a picture of a whole host of quarks and gluons, all mixed up together like beans in a bag. All “rushing around as fast as possible, at nearly the speed of light”. But in low-energy proton-antiproton annihilation, do we see all these quarks and gluons come spilling out? No. What we see, is light:

Image credit CSIRO, see The Big Bang & the Standard Model of the Universe

This is sometimes described as "two 'valence' up quarks and one 'valence' down quark, surrounded by a huge 'sea' of quark-antiquark pairs". In my opinion, this description is also misleading, because it implies that there are three specific "valence" quarks that are "real", while the other "sea" quarks are just "virtual" and physically distinct from the valence quarks. In fact, every individual quark is physically on the exact same footing. In particular, I believe that all of the quarks gain a mass contribution from the Higgs mechanism.

I share your sentiment that this description is misleading. But I will also say this: if all those zillions of quarks are physically on the exact same footing, why have you never ever seen a single one? What else do you think might be "rushing around" at the speed of light? 

As Prof. Strassler explains in the link above, the proton mass is best thought of as arising from the sum of three contributing terms:
  1. The sum of the rest energies of all of the (many!) quarks and antiquarks
  2. The kinetic energy of the quarks, antiquarks, and gluons
  3. The binding energy stored in the gluon fields, which is negative and actually decreases the proton mass relative to what it would be if the quarks did not interact.

Note this in the Wikipedia gluon article: "as opposed to virtual ones found in ordinary hadrons". The gluons in a proton are virtual. As in not real. Also note that it's the wave nature of matter, so check out what Martin van der Mark said in on the nature of stuff and the hierarchy of the forces: smaller mass means bigger wavelength, so you can’t fit a longer-wavelength 2.2 MeV quark inside a smaller-wavelength 938.27 MeV proton. Last but not least, remember what Einstein said: the mass of a body is a measure of its energy-content. 

I believe that the Higgs mechanism is responsible for the contribution (1), which is much larger than the naive 9.1 MeV from just three quarks, and therefore much more than 1% of the proton mass. Is this correct?

I think not. 

Also, do we have any kind of quantitative estimate of the contribution of each of these three terms? The number of up quarks and the number of up antiquarks are individually indefinite, but can we numerically estimate their ground-state expectation values in order to estimate (1)? I know that QCD is strongly enough interacting that it's not particularly useful to think in terms of individual quarks, but I'm curious whether anyone's attempted that calculation.

Sorry, I can't answer this part of your question. But I share your view that the 1% figure is wrong. Only I don't think it's too low. I think it's too high. 
