If a can be negative, then there is an easy counterexample: a = -n^2 for some concrete n > 1.
Suppose all numbers involved are positive. Using the Fundamental Theorem of Arithmetic, it helps to think about a number as a multiset of its prime divisors. Then m divides n iff the multiset of prime divisors of m is a subset of the corresponding set for n.
Suppose that n = p * q for some different prime numbers p and q, then n^2 = p * p * q * q. If a divides n^2 but does not divide n, then a must have two p's or two q's, say, a = p * p. Can you think of an example of p and q such that a <= n?