If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$ I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms.  So, is the following statement true?  
Claim:  "If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$."
Where $\mathbb{Z_n} \oplus \mathbb{Z_n}$ is the external direct product of $\mathbb{Z_n}$ and $\mathbb{Z_n}$.
I've been thinking about this for a few hours, but really can't figure out a good place to start (or a counter-example).  

First, $\mathbb{Z}_n \oplus \mathbb{Z}_n$ is abelian, while there are many non-cyclic groups that are non-abelian (take $S_3$ for example), so the answer to your question as written is immediately no.
However, what if we only consider abelian non-cyclic groups? Then $\mathbb{Z}_2 \oplus \mathbb{Z}_6$ and $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ are two counterexamples you might consider. [After OP's edit: a counterexample is $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$, which has order $16$ but is not $\mathbb{Z}_4 \oplus \mathbb{Z}_4$.]
[Note: this is a relevant result that you might already know: if $m$ and $n$ are coprime, then $\mathbb{Z}_m \oplus \mathbb{Z}_n \cong \mathbb{Z}_{mn}$.]
What you might then ask is if every abelian group can be written as the direct product of cyclic groups, and this is true, but not obvious: Classification of finitely generated abelian groups.
