Sorry for the clutter. I forgot a counterexample. Yes, it is
possible for (X*X)*X to have a final coalgebra but not X*X. On the
discrete category with two objects, A and B, let the bifunctor be
defined by:
* | A B
--+------
A | B A
B | A A
The unique cubical coalgebra is A but there is no square coalgebra.
What I don't have is a counterexample with an _associative_ bifunctor.