By definition, a Zariski closed subset of $\operatorname{Spec}A$ is a set of the form $V(I) = \{P \in \operatorname{Spec}A \mid I \subset P \}$. What if we work in a ZF model where AC is violated? (See below the note for the question)
Note: previous edits were confused by my troubles with the terms closed point and generic point, so I removed them. Sorry :(
$\mathrm{Edit}^2$: What a non-closed point that does not contain closed points looks like? Or, in non-Zariski language, if a ring $A$ has a proper ideal $I$ that is not contained in any maximal ideal, what does $\operatorname{Spec} A$ look like near the corresponding point?