We perform numerically exact calculations of quantum dynamics for a two-state molecular model system interacting with initially single-mode strongly amplitude- and strongly phase-squeezed fields (average photon number [(n) over cap] = 8). The second-order correlation function for these fields are taken to be 2, which is equivalent to that for the chaotic field with the identical average photon number. Although the responses of the molecule for these two squeezed fields are relatively similar to those for the chaotic field as shown in a previous paper, there are found to be some significant differences among these fields in the dynamical behavior of quantum-phase distributions of photons, i.e., the Pegg-Barnett phase and the Q function distributions. These differences are also elucidated by the dynamics of the information entropy for the molecule, i.e., molecular entropy, which represents the degree of entanglement between the molecule and the photon field.