This is the question on Futility Closet
Is there really no answer?
According to Wikipedia (inscribed square problem), this is an open problem - that is, we don't currently know the answer. It's known to be true for "nice enough" curves (Stromquist's Theorem).
So it is true for e.g. all piecewise smooth curves, which are the kind we tend to imagine. Stromquist more generally implies an affirmative answer for curves that are piecewise graphs (as in $y=f(x)$) of continuous functions. Also see this page for further discussion.