(This question is related to this one)

Let $k$ be a field and consider the category $Sch/k$ of schemes over $k$, say also separable and of finite type. The Yoneda embedding $$ Y:Sch/k \to Pre(Sch/k) $$ does not respect colimits but if you factorize $Y$ through $$ Y:Sch/k \to Shv(Sch/k) $$ with respect to the Zariski Grothendieck topology (given by the Zariski open immersions), it does respect some colimits (if you want you may replace $Sch/k$ by the category of commutative algebras over $k$ of finite type).

In particular a pushout of the form
$$
U~\xleftarrow{f}~ U\cap V ~\xrightarrow{g}~ V
$$
in $Sch/k$ where $f$ and $g$ are **open** immersions is also a pushout in $Shv(Sch/k)$.

The pushout of two **closed** immersions
$$
B\leftarrow A \rightarrow C
$$
in $Sch/k$ exists in general but let's consider the situation where $A,B,C$ are affine because in this case the existence is 'immediate' (whatever that is) by the antiequivalence to $k$-algebras. For example the coordinate cross $Spec k[X,Y]/(XY)$ is the pushout of $ \mathbb{A^1}\leftarrow Spec k\to \mathbb{A^1}$.

My question is

What happens to these 'closed' pushouts under the Yoneda embedding into Shv(Sch/k)? Are they preserved or is there an affine counterexample?

**Edit:** What happens if one takes sheaves with respect to the etale topology?