We discuss the appearance of supersolid phases for interacting hardcore bosons on the square lattice when, in addition to the standard nearest neighbor hopping and repulsion, correlated or next-nearest neighbor hopping is present. Having in mind dimer-based quantum magnets in a field described by effective bosonic models of this kind, we put special emphasis on a comparison between the different cases of relative signs of the kinetic processes, which correspond to unfrustrated or frustrated magnetic models. In the unfrustrated case, we compare Quantum Monte Carlo simulations with a mean-field (classical) approach, which is shown to give qualitatively correct results. Using this classical approach for the frustrated case, we find that, the phase diagram is generically richer than in the unfrustrated case. We also investigate in detail the differences between standard next-nearest neighbour and correlated hopping over the diagonal, with the conclusion that both cases are similar if checkerboard order is present at half-filling, while a supersolid phase can be stabilized without any adjacent solid phase only in the case of correlated hopping.