Neutron stars are the most compact form of matter that we know about; they’re even denser than the nucleus of an atom.

Neutrons, protons and electrons are fermions, meaning they must obey the Pauli exclusion principle. No two neutrons (or protons or electrons) can be in the same quantum mechanical state. If you take ordinary star matter (plasma made of dissociated protons and electrons) and squeeze it, eventually the electrons will nearly overlap in their states. You’d have two electrons with nearly the same energy, spin and location. They cannot overlap though, so this creates a repulsive force that prevents the matter from further compression; this is called the electron degeneracy pressure.

If the compressive force overcomes this pressure, then the electrons can capture on the protons to form neutrons. Neutrons and protons also have degeneracy pressures, but they can be packed much more densely than electrons. This is because their wavelength is shorter. The wavelength of a massive particle is inversely proportional to its mass, and protons and electrons are about 2000 times the mass of electrons. So compressed ordinary matter will inevitably become pure neutrons, simply because this is the most compact form.

A pure electron or pure proton star wouldn’t be as compact because both are charged particles so there would be Coulomb repulsion (this isn’t an issue in ordinary matter since the number of electrons and protons is roughly equal). You’d also need to somehow separate the electrons from the protons, and this isn’t a process that would naturally occur in a collapsing star.

Because neutrons are what you get when an elecron and a proton love each other very very much...

So basically if you sqeeze anything (made of atoms) hard enough you get neutrons.

It's not like it's "impossible" to sqeeze a bunch of protons, but stuff is usually made of atoms, not of protons, hence you don't get such a weird thing as a giant segregation of protons just lying there.

Charge is (basically) conserved, so if you start with a non-charged star you need a non-charged result. All ordinary matter is close to balanced in charge. (Even things on earth with a really high voltage)

As for why not a mix of both, neutrons are just more compact. Literally, as the start collapses, the massive pressure "squishes" protons and electrons together, and they undergo the reverse of a radioactive decay. It's just because that's more energetically favourable than taking up the extra space, and a neutron results.

Probably the simplest answer is protons and electrons repel themselves with a force far stronger than gravity. Neutrons don’t repel themselves in that way.

It's down to the way they are formed. Basically, once gravity starts winning in the internal forces within a star, it collapses on itself. At that point, it's made up of protons, neutrons and electrons, like most visible matter. However, if the star is large enough, the force from the gravity compresses the atoms so much that the protons and electrons get forced together. And the tl;dr version is that when this happens, their charge cancels out, and you get more neutrons.

There's actually a great XKCD "What If?" post that covers this: https://what-if.xkcd.com/140/

I'm starting here: https://physics.stackexchange.com/questions/116921/can-there-be-electron-and-or-proton-stars

I don't have the physics chops to understand most of what is being said, but if I can bang rocks together for a moment it sounds like it would fly apart, or, if you forced it all together somehow they would literally become neutrons.