There is an explicit counterexample of this statement in $\mathrm{Top}$, but the statement is true in the categories $\mathrm{Top}_\mathrm{conv}$ of convenient topological spaces, $\mathrm{Set}$, $\mathrm{Ab}$, $\mathrm{Grp}$, and $\mathrm{Vec}_k$ of vector spaces over a field $k$ (and more generally in the category $\mathrm{Mod}_R$ of $R$-modules over a ring $R$).
First for the proofs that the statement holds in the latter categories. For $\mathrm{Top}_\mathrm{conv}$ and $\mathrm{Set}$ (and more generally any cartesian closed category), it holds because these categories are cartesian closed, so $-\times E$ preserves colimits because it is a left adjoint. For $\mathrm{Ab}$ and $\mathrm{Mod}_R$ (and any additive category), this follows as these categories have biproducts: products and coproducts agree via the canonical map. Now suddenly the question whether $-\times E$ preserves pushouts becomes the question whether $-\sqcup E$ preserves pushouts, and this is true as colimits commute with colimits. Finally, for $\mathrm{Grp}$ we need to work a little bit. The category $\mathrm{Grpd}$ of groupoids is cartesian closed, and the inclusion $\mathrm{Grp}\to\mathrm{Grpd}$ of groups as one-object groupoids preserves pushouts and products. Namely, $\mathrm{Grp}$ is embedded as full subcategory, so any limit or colimit of one-object groupoids that results in a one-object groupoid also computes that limit or colimit in groups. The functor $\mathrm{ob}\colon\mathrm{Grpd}\to\mathrm{Set}$ that takes the set of objects of a groupoid has both adjoints (exercise), so the set of objects of a limit or colimit of groupoids is computed as the limit or colimit of the underlying sets of objects of the groupoids involved. The product of singleton sets or the pushout of a diagram of singleton sets both result in another singleton set, and this proves that $\mathrm{Grp}\to\mathrm{Grpd}$ preserves products and pushouts (more generally, this argument proves it preserves all limits and all connected colimits). Cartesian closedness of $\mathrm{Grpd}$ therefore implies that $-\times E\colon\mathrm{Grp}\to\mathrm{Grp}$ preserves pushouts, and more generally all connected colimits.
Finally, for the counterexample in the category $\mathrm{Top}$ of all topological spaces, we follow this answer, and define $i\colon\mathbb{Z}\to\mathbb{R},n\mapsto n$ and $j\colon\mathbb{Z}\to\mathbb{R},n\mapsto n+1$. Then Oscar Cunningham shows in the linked answer that the natural map
$$\mathrm{coeq}(i\times\mathbb{Q},j\times\mathbb{Q})\to\mathrm{coeq}(i,j)\times\mathbb{Q}$$
is not a homeomorphism. We have a pushout square
$$\require{AMScd}
\begin{CD}
\mathbb{Z}\sqcup\mathbb{R}@>{(i,\mathrm{id})}>>\mathbb{R}\\
@V{(j,\mathrm{id})}VV @VVV\\
\mathbb{R} @>>> \mathrm{coeq}(i,j)
\end{CD}
$$
and another pushout square
$$\require{AMScd}
\begin{CD}
\mathbb{Z}\times\mathbb{Q}\sqcup\mathbb{R}\times\mathbb{Q}@>{(i\times\mathbb{Q},\mathrm{id})}>>\mathbb{R}\times\mathbb{Q}\\
@V{(j\times\mathbb{Q},\mathrm{id})}VV @VVV\\
\mathbb{R}\times\mathbb{Q} @>>> \mathrm{coeq}(i\times\mathbb{Q},j\times\mathbb{Q})
\end{CD}
$$
Since $\mathbb{Z}\times\mathbb{Q}\sqcup\mathbb{R}\times\mathbb{Q}\cong(\mathbb{Z}\sqcup\mathbb{R})\times\mathbb{Q}$, we find a pushout square
$$\require{AMScd}
\begin{CD}
(\mathbb{Z}\sqcup\mathbb{R})\times\mathbb{Q}@>{(i,\mathrm{id})\times\mathbb{Q}}>>\mathbb{R}\times\mathbb{Q}\\
@V{(j,\mathrm{id})\times\mathbb{Q}}VV @VVV\\
\mathbb{R}\times\mathbb{Q} @>>> \mathrm{coeq}(i\times\mathbb{Q},j\times\mathbb{Q})
\end{CD}
$$
Because of the above non-homeomorphism, the natural diagram
$$\require{AMScd}
\begin{CD}
(\mathbb{Z}\sqcup\mathbb{R})\times\mathbb{Q}@>{(i,\mathrm{id})\times\mathbb{Q}}>>\mathbb{R}\times\mathbb{Q}\\
@V{(j,\mathrm{id})\times\mathbb{Q}}VV @VVV\\
\mathbb{R}\times\mathbb{Q} @>>> \mathrm{coeq}(i,j)\times\mathbb{Q}
\end{CD}
$$
is not a pushout square, and this the counterexample we were after.
