The outward pressure is neutron degeneracy pressure. Electron degeneracy pressure is what stops white dwarfs collapsing in but when this is less than the gravitational pressure, inverse beta decay creates neutrons. Neutron stars generally have a radius of (electron mass/proton mass)*white dwarf radius which turns out to be tiny, something like 10km!
Collapse is nothing to do with heat from fusion. Even if the heat from a stable neutron star had run out and it reached a few K, it would not collapse because the degeneracy pressure is not a function of temperature. The only way to make a stable neutron star tip over the edge to a black hole is by accretion from another object like a second star in a binary system.