Multiplication via squaring and addition

Per your comment, the precise question you're asking is:

Is multiplication definable in the structure $(\mathbb{N}; +,\cdot^2)$?

The answer is yes: we have $z=x\cdot y$ iff $z+z+x^2+y^2=(x+y)^2$.

This is a bit unsatisfying; can we do better?

Well, one natural hope would be for a specific term built out of $+$ and $\cdot^2$ which gives multiplication. E.g. raising to the fourth power isn't just definable, it's given by the term $(x^2)^2$. So we now ask:

Is the term $xy$ equivalent (in the obvious sense) to a term in the language $+,\cdot^2$?

The answer to this new question is no. One way to see this is by taking derivatives. Suppose $t(x,y)$ is a term built out of $+$ and $\cdot^2$. Then when we write ${\partial\over\partial x}t(x,y)$ as a fully-cancelled-where-possible sum of monomials, every monomial in which $y$ occurs have even coefficient$^1$. But the monomial $xy$ itself doesn't have this property.$^2$

$^1$This takes proof, but it's a straightforward induction so I'll leave it to the reader.

$^2$OK fine, technically we need to prove that the fully-cancelled-sum-of-monomials form of a polynomial is unique, but meh - I'll leave it to the reader as well. Induction builds character.