The anti-particle corresponding to a proton or an electron is a particle with an equal mass, but an opposite charge. So what is the anti-particle corresponding to a neutron (which does not possess a charge)? And if it is just another neutron, will its collision with the original neutron be as destructive as the collision of a proton with an anti-proton or an electron with an anti-electron?
The anti-particle corresponding to a neutron is an anti neutron!
The neutron is made up of one up quark and two down quarks. The anti-neutron is made up of an anti-up quark and two anti-down quarks. Both have zero charge because the charges of the quarks within them balance out.
You are correct that elementary particles with no charge are often their own anti-particles. These tend to be vector bosons; for example the photon and the Z boson are their own anti-particles. The W$^-$ and W$^+$ are each other's anti-particles. It's a bit more complicated with the gluons because they carry a colour charge.
Amongst the fermions there are no particles known that are their own anti-particles. If such particles exist they would obey the Majorana equation and these theoretical particles are known as Majorana fermions.