Here are the details of Thierry's argument why the interval cannot be trivially fibrant.
Without loss of generality, we may assume that an empty system of size zero gets filled to constantly 0. By uniformity with respect to degeneracies, it follows that any empty system gets filled to constantly 0. Now consider a filling problem for a square with dimensions i, j with corner (0, 1) having specified value 1. Let phi denote the filler. By uniformity with respect to diagonals, since there is an empty diagonal, phi[i=j] = 0 since that diagonal is empty. By examining the DNF of phi, it follows that phi = 0 this can only be the case if phi is 0. This contradicts the system.
A similar argument can be made for composition by considering a cube with dimensions i, j and monotone filling direction k with bottom face having value 0 and edges where (i, j) is (0, 1) or (1, 0) having value 0 and k, respectively. Again, only uniformity with respect to degeneracies and diagonals is used.
So the interval is indeed an example of a non-uniformly trivially fibrant cubical set that is not fibrant.