Number of nucleons is conserved. You have to have the same number of protons and the same number of neutrons on both sides of the reaction equation. This is not different from balancing chemical reaction (just then you conserve number of atoms and charge, which is actually applying the same conservation principles).
Yes and no. Yes - you are right about accounting, number of neutrons after the reaction must equal number of neutrons after the reaction. No - reaction was initiated by an additional neutron, so you should take it into account as well, plus, in such reactions free neutrons are produced (and they further break more nuclei, which is why we call it a chain reaction), and these have to be accounted for too.