From what I understand, the neutrons themselves aren’t being compressed so much as the space between them gets smaller. The neutrons don’t physically push against each other, they resist the compression with something called degeneracy pressure. They can’t occupy the same space, so they will change states before that happens, which creates this pressure that resists the gravitational pressure.
You are correct.
But will electron degeneracy pressure will halt the gravitational collapse of a star if its mass is below the Chandrasekhar Limit?. A star exceeding this limit and without usable nuclear fuel will continue to collapse to form either a neutron star or black hole, because the degeneracy pressure provided by the electrons is weaker than the inward pull of gravity.