QED and Newton's 3rd law According to some lecture notes on propagators (see bottom of page) the Feynman particle/anti-particle hypothesis states:

The emission (absorption) of an antiparticle of 4-momentum $p^\mu$ is physically equivalent to the absorption (emission) of an particle of 4-momentum $-p^\mu$.

The antiparticle of a photon is a photon.
So if an electron emits a photon with 4-momentum $p^\mu$ that is equivalent to it absorbing a photon with 4-momentum $-p^\mu$.
Is that right?
So is this the origin of Newton's 3rd law of action and reaction?
 A: Excellent question. There are forces for which a force-carrier particle is not it's own antiparticle (eg: Strong force or the Weak force) so if we accept your explanation, then we might have to abandon Newton's 3rd law for those forces, which is implausible.
Edit: Maybe I'm being obtuse. I don't think really matters that the antiparticle is not the same particle, so long as it can be interpreted as some particle with the opposite momentum. We never care about what kind of particles carry the force -- if there are many, we just sum over them anyways. So, maybe the reason that you gave does get to the heart of the matter.
Also, in my explanation below, the derivation of the potential from the propagator depends on the Feynman pole prescription which is the interpretation that every particle has a corresponding antiparticle. So I think you hit the nail on the head :-)

If I were to take a shot, I would say that the law "comes from" the fact that only the relative position matters, when two objects exerting a force on each other.
Think of the energy in the system and the force as $- \frac{\partial U}{\partial x}$. The "potential" is obtained by inverting the propagator must depend only on the distance between the charges and nothing else (by rotational invariance). Distance $= |x_1 - x_2|$. Physically, you can think of either charge in the potential of the other -- so the energy must be symmetric in their charges. This will give us a force law similar to what we've seen before
$$U (q_1, x_1 ; q_2, x_2) \sim \frac{q_1 q_2}{|x_1 - x_2|} e^{- m |x_1 - x_2|}$$
where $m$ is the mass of the force-carrier particle. For a photon or graviton $m=0$ so we get the usual Coulomb/Newtonian potential.
Once we have this expression for the energy, Newton's 3rd law is purely a result in classical mechanics. By computing $F_1 = -\frac{\partial U}{\partial x_1}$ and $F_2 = -\frac{\partial U}{\partial x_2} = - F_1$ we can show that the forces on the two charges are equal and opposite.
