Let $X$ be a completely regular, Hausdorff topological space and let $\cal F$ be a $z$-ultrafilter on $X$. Then for each zero set $W$ in $X$, either $W\in \cal F$ or there exists $Z\in \cal F$ such that $Z$ does not meet $W$ (this is the $z$-ultrafilter property). Now suppose that $X$ is additionally normal. Then is it true that for every closed set $W$ in $X$ either $W$ contains an element $Z$ of $\cal F$ or there exists $Z\in \cal F$ such that $Z$ does not meet $W$?
No: think of what happens with $\omega_1$ in the usual topology. This is certainly normal (even hereditarily normal), and since every real-valued continuous function on $\omega_1$ is eventually constant, the co-bounded sets form a $z$-ultrafilter. Now let $W$ be the set of countable limit ordinals.