[Slightly too long for a comment, so I post it community wiki answer.]

The kernel of the epimorphism $\quad\varphi : G\times G \to H\times H\quad$ is a normal subgroup of $G\times G$, for which by an [easy calculation](https://mathoverflow.net/questions/23692/what-are-the-normal-subgroups-of-a-direct-product/23722#23722) one can show that

$$N_{-}:=[\pi_1(N), G]\times [\pi_2(N), G] \le N \le \pi_1(N)\times \pi_2(N)$$

with $\pi_i$ the projection on the $i$-th coordinate. As $G$ acts trivially on $\pi_i(N)/[\pi_i(N), G]\;$, $N/N_{-}$ is central in $(G\times G)/N_{-}\;$. [This might be the motivation for Yves' second comment.]

Similar statements hold for the preimages $\varphi^{-1}(H \times 1)$ and $\varphi^{-1}(1 \times H)$ , one can also play around with [Goursat's lemma](http://en.wikipedia.org/wiki/Goursat_lemma), but I'm still undecided if I should rather try to prove or disprove the question.