We can cut any square on $n$ squares if $n>5$ and $n=4$.
The proof is easy by induction. Base cases $n=6,7,8$ are easy to find and then since we can cut a square on $4$ squares we get $3$ new squares, so we go from $n\to n+3$ and we are done.
But I can not find a proof that we can't cut it on $5$ squares. I suppose we should search for some contradiction, but...?