If the sun and stars are supported by gas pressure, what supports a neutron star?
Neutron degeneracy pressure.
Protons, electrons, and neutrons make up all of the elements on the periodic table. They are also examples of fermions, or particles with half integer spin. All fermions obey a principle of quantum mechanics called the Pauli exclusion principle, which states that no two identical fermions may occupy the same space at the same time.
In degenerate stars, such as white dwarfs or neutron stars, the Pauli exclusion principle generates a degeneracy pressure which supports the star against further collapse. In essence, the electrons cannot physically be compressed any further.
If, however, the gravity of the star creates enough pressure to overcome electron degeneracy pressure, the star will continue to collapse until neutron degeneracy stops it. If gravity overcomes neutron degeneracy pressure, than the star will continue to collapse into a black hole.