Step 1: Translate the statements to sets.
All Z are Y means $Z \subseteq Y$; No Y is X means $Y \cap X = \varnothing$; Every X is W means $X \subseteq W$.
Step 2: Combine the first two facts.
Since Z lives entirely inside Y, and Y shares nothing with X, Z also shares nothing with X.
Step 3: Evaluate Conclusion II.
That gives $Z \cap X = \varnothing$, i.e. Z are not X, which therefore definitely follows.
Step 4: Examine Conclusion I.
Conclusion I claims Some W are Z. We know X is inside W, but Z need not touch W at all.
Step 5: Show I can fail.
One can place Z and W as disjoint regions, with only X forced inside W, so no overlap of W and Z is guaranteed; Conclusion I does not necessarily follow.
Step 6: Pick the option.
Only Conclusion II follows.
\[ \boxed{\text{Only Conclusion II follows.}} \]