[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 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, but I'm still undecided if I should rather try to prove or disprove the question.