Why was this question asking non-trivial examples of closed subschemes closed?
Your question was: "I would like to know examples of closed subschemes of $X$ other than $Z_n$", and when examples were given in the comments, you rejected them as trivial. Rather than explaining yourself further, e.g., describing why the examples were trivial, you repeatedly asked whether abx knows "what $Z_n$ means". You changed the question to include the condition that the underlying subspace is $Z$, but rejected Moret-Bailly's suggested answer as trivial.
Despite all of the comments, you still haven't explained why the non-projective property of the variety makes the question non-trivial (or rather, what it is about the projective property makes the question trivial). It is not constructive to declare that something is trivial without explaining why!
Based on this behavior, I think it is reasonable to conclude that you are unlikely to refine the question to the point where it admits a precise answer in the near future.