Could dark energy just be particles with negative mass? The title speaks for itself.
Dark matter: We see extra attractive force, and we posit that there are particles which create such a force, and use the measure of that force to guess their locations.
Dark Energy: We see extra repulsive force.
Only thing is, dark energy is uniform. So I suppose the stuff would have to be (at least somewhat) uniformly distributed throughout the universe. How uniform do we know it to be? Could the "stuff" be somehow a part of empty space itself?
 A: Matter and dark matter are also evenly distributed throughout the observable universe, at least on the largest scales. What makes dark energy different isn't that it is uniformly distributed, but that it has a constant density. The amount of dark energy per cubic meter of universe is the same regardless of the total volume of the universe. If the universe is twice as big, there is twice as much dark energy, so the universe expands twice as fast. Hence the accelerated expansion. This must be a property of space itself. If these were particles they would simply dilute away as volume increased.
A: “If dark energy would consist of particles, it would dilute with the growing radius of the universe to the third power, since the total number of particles would stay the same while the volume increases. What observations found was that dark energy rather behaves like a constant which does not thin out, that's why it is also known as the cosmological constant. That means even if the universe expands, the amount of dark energy per cubic meter stays (at least approximately) the same.”
The answer above was conventionally correct. Modern science has now developed further since then. There was a result last year by an Oxford professor Jamie Farnes, widely discussed in international media, about possible creation of negative mass particles.
https://arxiv.org/abs/1712.07962
When negative mass particles are created by a creation field, they do not thin out and can behave as a cosmological constant. Apparently the idea is also testable.
A: If dark energy would consist of particles, it would dilute with the growing radius of the universe to the third power, since the total number of particles would stay the same while the volume increases. What observations found was that dark energy rather behaves like a constant which does not thin out, that's why it is also known as the cosmological constant. That means even if the universe expands, the amount of dark energy per cubic meter stays (at least approximately) the same.
A: Solutions to this problem exist in peer-reviewed bimetric theories (many papers available, just search "negative mass in cosmology" on the internet) which separate positive energy from negative energy species in two conjugate pseudo-Riemannian metrics $g_{\mu \nu}^{(+)}$ and $g_{\mu \nu}^{(-)}$ where distances between two points A and B in spacetime are different according to the metric where they are measured. 
The positive and negative sectors have different space scale factors $a^{(+)}$ and $a^{(-)}$ that you can approximate to the measured "radius" giving the "size" of the universe. As the positive sector and the negative sector are both expanding but are also in mutual repulsion and in unstable equilibrium from the Big Bang, one of the sectors quickly passed in front of the other. In a simplified way, imagine these two sectors as two concentric bubbles expanding together, being also in mutual repulsion. Hence the bubble ahead is accelerating, hassled by the one which stayed behind, which itself undergoes a decelerating expansion for the same reason. 
Calculations show that the inner bubble will always stay very contracted and its content (hot gas nebulae made of antiparticles of negative mass), although mutually gravitationally attracting, will never coalesce in conglomerates by gravitational instability, as the cooling time in this sector is equal to the age of the universe (thus no stars, no planets, no life in the contracted negative sector). Therefore, although the positive mass equals (in absolute value) the negative mass of the universe: (|+m| = |-m|), the contracted state of the negative sector (smaller space scale factor) makes the negative content of the universe appear as having a much greater mass density (|-ρ| ≫ |+ρ|). So it is the negative content of the universe, its negative energy density, which is in charge, which drives the observed accelerating cosmic expansion.
The cause of this accelerating expansion is what the mainstream ΛCDM model calls "dark energy" which would be the expression of a mysterious negative and exotic "repulsive power of the vacuum". According to bimetric theories of gravity (not to be confused with bigravity theories involving massive gravitons), dark energy is simply the repulsive pressure due to the overall negative energy density of the contracted negative sector, made of the cosmological antimatter at t<0 (see Sakharov conditions and CP violation and the matter–antimatter imbalance, 1967), having a negative energy and negative mass due to T-symmetry (see Symplectic geometry and dynamical group theory for reference, especially the complete Poincaré group and its relation to the arrow of time and energy inversion).
A: Density of Dark Energy is very low: $6.91 × 10^{−27} kg/m^3$
And, it permeates whole space uniformly (so it dominates the universe with 68% presence). As for your empty space claim, it's not true. Due to Uncertainty Principle in action, you can't measure energy precisely for a precise time interval.
A: Negative mass particles might dilute away, so no cosmological constant, unless they were products of a hypothetical compact star, exotic star, or a black dwarf star decaying: constantly spitting out unknown particles... that may be hitting us from the future (from the decay of a futuristic black dwarfs jettisoning an exotic particle like a tachyon, neutrino, or hawking radiation with such low mass that it is faster than light and traversing backward and or forward across time and space),
Compact exotic stars could generate a constant supply of short-lived micro-black holes to compensate for dark matter as well. Our dillema is we don't know a lot about particles. What particles might exist is the difference between a big chill and a big bounce. I mean, the cosmological constant's effects on physics us subject to change should particle physics cause a cosmology episode to occur.
A: Could Dark Energy just be particles with negative mass?   Yes!
What we are looking for in cosmology is one attraction term and one repulsion term. If a negative mass exists, we can get it.
If negative mass and positive mass coexist, gravitational potential energy consists of the following three items.
\begin{equation}
U_T = \sum\limits_{i <
j}^{} {( - \frac{{Gm_{ + i} m_{ + j} }}{{r_{ + + ij} }}) + \sum\limits_{i < j}^{} {( - \frac{{G( -
m_{ - i} )( - m_{ - j} )}}{{r_{ - - ij} }})} + \sum\limits_{i,j}^{} {( - \frac{{G( - m_{ - i} )m_{ + j}
}}{{r_{ - + ij} }})}}
\end{equation}
\begin{equation}
U_T = \sum\limits_{i < j}^{} {( -
\frac{{Gm_{ + i} m_{ + j} }}{{r_{ + + ij} }}) + \sum\limits_{i < j}^{} {( - \frac{{Gm_{ - i} m_{
- j} }}{{r_{ - - ij} }})} + \sum\limits_{i,j}^{} {( + \frac{{Gm_{ - i} m_{ + j} }}{{r_{
- + ij} }})} }
\end{equation}
\begin{equation}
U_T = U_{ + + } + U_{ - - } + U_{ - + }
\end{equation}
$U_{--}$ : attraction term
$U_{-+}$ : repulsion term
The present cosmological constant can be obtained by adding potential ${U_\Lambda }=-\frac{1}{6}\Lambda m{c^2}{r^2}$ to mechanical energy conservation equation. If we insert 'new potential energy term' into mechanical energy conservation equation, we will get a similar term such as $ - \frac{1}{3}\Lambda {c^2}$ term.
At this time, let's insert the above new gravitational potential energy term into it.
If $U_{++}$, $U_{--}$, $U_{-+}$ has a ratio(4.9% : 26.8% : 68.3%) between each other, maybe, we will estimate that ratio of energy density such as 4.9% : 26.8% : 68.3% exist.
This model can prove the energy composition(Matter : Dark matter : Dark energy) ratio of the universe and CCC (Cosmological Constant Coincident) Problem. 
Roughly calculation :
Average of WMAP and Planck - Matter : Dark Matter : Dark Energy = 4.75% : 25.05% : 70.20%
Average of Pair Creation Model - Matter : Dark Matter : Dark Energy = 4.75% : 25.00% : 70.25%
Even though the origin of dark energy is particles with negative mass, the phenomenon that dark energy increases as the universe expands can appear. How?
Because we judge the components of the universe through gravitational effects rather than mass energy. Please refer to the following my papers.
My Paper-1 : Pair Creation Model of the Universe From Positive and Negative Energy
https://www.researchgate.net/publication/275056453
My Paper-2: Dark Matter is Negative Mass
https://www.researchgate.net/publication/324525352
===========
Negative mass was found but ...
From the discovery of the accelerating expansion of the universe, people generally claim the existence of cosmological constants or vacuum energy. However, by borrowing their logic, the accelerating expansion of the universe can be interpreted as evidence for the existence of negative mass.
In the discovery of accelerating expansion of universe in 1998, negative mass and negative energy were the first result of the field equation and Friedmann equations. Since those who received the first result had the wrong stereotype of negative mass and negative energy, so they rather modified the field equation. They resurrected the cosmological constant and modified the equation.
Nobel lecture by Adam Riess
https://www.nobelprize.org/mediaplayer/index.php?id=1729
Refer to 10m : 50s ~
======

Negative Mass?
Actually the first indication of the discovery

Days later… What does this mean?
There cannot be negative mass, but would Einstein’s Cosmological Constant explain this acceleration?
=======
Let's remove his preoccupation and step back and look at the accelerating expansion of the universe. This is a phenomenon that can also be explained by the presence of negative masses.
The first findings were as follows:
HSS(The High-z Supernova Search) team : Ω_m = -0.38(±0.22) : negative mass density
https://arxiv.org/abs/astro-ph/9805201
P14 and Table 8
Thay have a sense of rejection of negative mass (energy), so they use logic to introduce negative pressure~
From the second Friedmann equation or acceleration equation,
$\frac{1}{R}\frac{{{d^2}R}}{{d{t^2}}} =  - \frac{{4\pi G}}{3}(\rho  + 3P)$
Since ρ is the energy density in the acceleration equation, the pressure term P also has a dimension of energy density(c=1). In the acceleration equation, without negative mass density, it is impossible to create acceleration expansion.
The mainstream produces acceleration expansion by setting the pressure of the cosmological constant or vacuum energy to P = -ρ.
The notion created by the mainstream has an inertial mass of + 1(ρ), with gravitational mass of -2(ρ). Not only different signs, but different values. It violates the principle of equivalence of inertial mass and gravitational mass, which is the basis of general relativity theory.
(-2ρ) = (ρ + 3P) = (ρ +3(-ρ)) = (-2ρ)
Their rejection of negative mass has created this trick or logic. If we accepted the concept of negative mass, we did not have to make these trick or logic.
We have to know that not the field equation has disposed the negative mass, but our stereotype disposed that negative mass.
And we have to know that rejection logic of negative mass, which is the root of stereotype, is wrong.
Moreover, we considered vacuum energy as the source of dark energy, but the current result of calculation shows difference of 10^120 times between the two values (observation and calculation), which is unprecedented even in the history of Physics.
The model also has a CCC(Cosmological Constant Coincidence) problem.
The notion created by the mainstream does not conserve energy.
Anyway, negative mass (energy) is a valid solution for accelerating expansion of the universe.
A: Despite what most answer says. Yes Negative energy particles can account for dark energy and dark matter. D will already gave a good exemple. Another article I would like to point out is this one:
https://www.researchgate.net/profile/Jean_Pierre_Petit/publication/277589002_The_Janus_Cosmological_Model/links/556de3d208aeab7772269cba.pdf
It is the Janus cosmological model. I copy the abstract for futur reference:

Abstract     :     The     Janus     Cosmological     Model     is     presented,     based     on     a     bimetric     approach
  fairly    different    from    usual    ones.    The    f
  irst    population    refers    to    ordinary    matter    while    the
  second     is     a     copy     of     that     last,     with     negative     mass     and     energy.     The     existence     of     such
  negative    component    is    justified    and    leads    to    good    fit    to    observational    data    especially    for
  the     acceleration     of     the     universe.     This     well     defined     single     kind     of     invisible     matter
  challenges    both    dark    matter    and    dark    energy,    with    respect    to    observational    data.

As you can see see. It accounts for the accelaration of the universe (Which mean the comoslogical constant is taken into account).
About your last question:

Only thing is, dark energy is uniform. So I suppose the stuff would have to be (at least somewhat) uniformly distributed throughout the universe. How uniform do we know it to be? Could the "stuff" be somehow a part of empty space itself?

In this model dark energy is not uniform. But it can still lead to the observed expension and accelration of the universe. The negative masses are mostly in the void around galaxy clusters.
