How about the positive integers with * as sum, with 1->3 as the only nonidentity arrow? The unique cubical coalgebra is 1 but there is no square coalgebra.
Oops, that gets associativity at the cost of * no longer being a functor (pointed out to me by Peter Selinger). Ok, let me dig myself in deeper by making my example more complicated. Instead of 1->3, put an arrow from i to j whenever i <= j <= 2i. Now every i is a square coalgebra but no i is a cubical coalgebra. Now adjoin a new object oo (infinity), with x+oo = oo+x = oo for all x, and the identity at oo as the only new arrow. oo is both a square *and* a cubical coalgebra. Since it is disconnected from the other square coalgebras there can't be a final such. But oo is the only cubical coalgebra, with only one self-map, making it a final cubical coalgebra. Vaughan