David Pincus in "A note on the cardinal factorial" (Fundamenta Mathematicae vol.98(1), pages 21-24(1978)) proves that $A^A=2^A$ does not imply the axiom of choice, therefore it does not characterise the sets for which $A=A\times A$. The counterexample is the model from his paper "Cardinal representatives", Israel journal of mathematics vol.18,pages 321-344 (1974). 

As Pincus writes on the last lines of [Pincus78]:

> c. Our arguments [ in the proof of "4.The Main Theorem; If $x$ is infinite then $2^x=x!$" ] have made little use of the particular definition of $x!$. Indeed let $\mathcal{F}$ be any set valued operation which satisfies: [...] 

> The statement "For every infinite $x$, $2^x=|\mathcal{F}(x)|.$", holds
> in $M$ (and therefore is not an equivalent to the axiom of choice).
> Examples of $\mathcal{F}$, apart from $x!$, are $x^x$ and $x^x-x$.

I should say that, after failing to come up with an answer myself, I found out about this by searching into my good old friend the "Consequences of the axiom of choice" by Howard and Rubin, Form 200, and Note 64. I try to reference this book whenever I can :)