The physical properties of the square water model, which is a generalization of the square ice to nonzero temperatures, is studied as a function of temperature and electric field. We determined the fraction of hydrogen bonds (HBs), the electric susceptibility, and the entropy of the model. We found that the usual independent-bond approximation gives poor predictions for the HB number when a polarization field is present. We compare the independent-bond results with Monte Carlo simulations, and with more accurate mean-field approximations obtained by the study of clusters of water molecules. At zero temperature, this model presents a first-order phase transition driven by the external electric field. The discontinuity in the HB number gives support to this behavior. We also obtained the exact partition function of the square water model in one dimension employing the transfer matrix technique. The zero field free energy in one dimension displays the same functional form on temperature as the one obtained in the two-dimensional version of the model via mean field approach.