Why do you think that the assumption is that n^2 is divisible by 7? And this is the assumption of what exactly?

Suppose you need to prove that for all n, P(n) implies Q(n). Here is a direct proof: fix an arbitrary n, assume P(n), use it to conclude Q(n). Here is a proof by contradiction: fix an arbitrary n, assume ~Q(n) (i.e., the negation of Q(n)), use it to conclude ~P(n).