$\DeclareMathOperator\Spec{Spec}$Does there exist a scheme $X/{\operatorname{Spec}(\mathbb{Z})}$ such that $\pi_1^\text{ét}(X)=\smash{\hat{\mathbb{Z}}}^2$?

It's well known that $\pi_1^\text{ét}(\Spec(\mathbb{F}_p))=\hat{\mathbb{Z}}$, for $p$ a prime. However, the scheme $$X_{p,q}:=\Spec(\mathbb{F}_p)\times_{\Spec(\mathbb{Z})}\Spec(\mathbb{F}_q)$$ is empty if $p\neq q$, and isomorphic to $\Spec(\mathbb{F}_p)$ if $p=q$, so neither work. The idea was to use the natural map, $$\pi_1^\text{ét}(X\times_{S}Y)\rightarrow \pi_1^\text{ét}(X)\times \pi_1^\text{ét}(Y)$$ somehow. So does such a scheme exist?