Is this a valid z-transform of $ \frac{1}{n^2}u[n] $?
What you've got is definitely the z-transform of that signal (by definition).  For a more compact expression, notice that the polylogarithm is exactly the infinite sum you've shown (just replace $ z \rightarrow 1/z$ in the Mathworld expression). Unfortunately I don't think there's any nicer way to write it; sometimes there just isn't.