I can offer a reason why the neutron is chosen over the proton, and one for why not something like an electron (or other lepton). See below...
First, why not a proton? The proton belongs to the same particle class as the neutron - both are hadrons, particles composed of quarks. However, because our target (U-235) carries a large positive charge, the proton bullet would be strongly repelled by the electric force, making it very difficult to produce a collision in which the proton was absorbed into the nucleus.
(The same could be said of any particle such as an alpha-particle, which is itself a small nucleus.)
The neutron, on the other hand is neutral, which means the electromagnetic force plays little role in preventing the collision, and the neutron will reach the nucleus at a lower energy. (Indeed, the neutron must not have too much kinetic energy, or it could pass through the nucleus, without being absorbed.
In the case of an electron, a big problem is that this particle does not experience the strong nuclear force, (nor does any other lepton) and there would be no interaction with the nucleus that could result in fission.