Earlier quoted context omitted.
Here's a thought experiment that shows that there can't be electrons: "If you had an object with negative charge, then putting that object near a positive charge would cause it to accelerate towards the positive charge." Getting closer increases the force. "This is a positive feedback loop that immediately results in the object and the positive charge exerting infinite amounts of force on each other, because" getting…
So, electrons are attracted to protons by the electromagnetic force, but they don't make it all the way there because they are repelled by some other force. You don't have a paradox because our model of the electromagnetic force doesn't specify that it's the only force there is. But if a negative mass is moving towards you and encounters a repulsive force, that repulsive force will, by definition, move the negative m…
A repulsive force is not the best way to think about it. The potential of the nucleus is the usual -1/r, and goes to (negative) infinity at zero. A repulsive force would be incorporated into the potential and appear as a bump around the nucleus, and would mess up the electron orbital.
A hand-wavy explanation of why the electron doesn't fall in: if you try to push the electrons into the nucleus, you necessarily localize the electron into a smaller volume; this means its wavefunction must get "spikier" and therefore it has more kinetic energy. This kinetic energy rises faster than the potential energy drops, so the state of lowest energy is actually found at an average radius > 0.
Note that it took quantum mechanics to rescue the atom. A classical electron could fall into the nucleus.
I do not see any problem with a negative-mass particle accelerating towards the force. The analogy with opposite charges seems right to me.