If any object could become a black hole, could any object become a neutron star? A black hole doesn't necessarily need to form from a star; theoretically, it could form from any extremely dense object. In fact, many astronomers differentiate certain black holes, like supermassive ones, from stellar ones (ones that form from stars).

However, could the same apply for neutron stars? Neutron stars only form because of the intense gravity during a star's collapse: electron capture is forced to happen, and the majority of the star becomes neutrons. Could this potentially happen to non-stellar objects, if the gravity forces electron capture?

If so, why don't we see as many of these "neutron objects" as we do non-stellar black holes?
There can be no such thing as a &quot;supermassive neutron star&quot;. The theoretical upper mass limit for a neutron star is somewhere between 2.2 and 3 solar masses. Any more massive and they inevitably form black holes. So I am not clear what kind of &quot;neutron objects&quot; you were thinking of?
Nor is it clear what you mean by &quot;non-stellar&quot; objects that will have the densities required to make neutron-degenerate matter? There aren't any apart from (I) the cores of massive stars at the ends of their lives. (II) Massive white dwarfs if they accrete matter over and above their Chandrasekhar limit. Furthermore, there is a minimum mass for a neutron star. Although the observed lower limit to those seen in nature is (so far) around $1.15 M_{\odot}$, there is a theoretical lower limit of about $0.1$ to $0.2M_{\odot}$ set by the stability of the neutron against decay in self-gravitating material ( See https://physics.stackexchange.com/questions/143166/what-is-the-theoretical-lower-mass-limit-for-a-gravitationally-stable-neutron-st/143174#143174 ).
It is true if you could arrange to compress any matter to densities above $\sim 10^{15}$ kg/m$^3$ it would form neutron-degenerate material. But this requires (as far as we know) the conditions I listed above.