My textbook states "To show a theorem is false, it is enough to find a single case where the implication does not hold. However to show that a theorem is true, we must give a proof that covers all cases".

Then it asks "There exists a positive integer n such that $\displaystyle n^2<n$ Would it make sense to attempt to prove this statement with a counter example?" The answer is no you need to prove it for all n.

But it just said you can prove a theorem false with a counter example?