Is there a space $X$ that is not locally compact, and a compactification $Y\neq \beta X$ of $X$, such that $\text{cl}_Y Z_1 \cap \text{cl}_Y Z_2$ is finite whenever $Z_1$ and $Z_2$ are disjoint zero sets in $X$?
If we allow locally compact spaces, we could of course just take $X=\omega$ and let $Y$ be its one-point compactification.
Let $X$ be any non-locally compact completely regular space. Take $\beta X$ and identify together two points of $\beta X\setminus X$ to get a quotient $Y$ of $\beta X$ which is still a compactification of $X$ (since $X$ is not locally compact, $\beta X\setminus X$ has more than one point). You can compute the closure of a set $Z\subseteq X$ in $Y$ by taking its closure in $\beta X$ and then projecting to $Y$ by the quotient map, so it follows that disjoint zero sets have closures which are disjoint except possibly for the point of $Y$ corresponding to the two points of $\beta X$ we glued together.