Recently, Zhao et al. [Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 2684] reported the phase behavior of monolayers of polymeric Brownian squares platelets. As the density is increased, this system exhibits the formation of two crystal structures not expected for particles with square symmetry, namely, a hexagonal rotator crystal phase and a rhombic crystal phase. Molecular simulations by Wojciechowski and Frenkel [Comp. Met. Sci. Technol., 2004 10, 235] had predicted instead the formation of tetratic and square crystal phases. In this work, we report Monte Carlo simulation results of rounded hard squares of varying degrees of roundness, hence interpolating between disks and perfect squares. Our simulations show that the roundness of the particles gives rise to the phases observed by Zhao et al. and further provide a roadmap for the regions of stability of different ordered phases as a function of particle roundness. In particular, our results suggest that depending on the degree of roundness, the isotropic phase would transition either into a hexagonal rotator phase through a hexatic-like intermediate, into a square phase through a tetratic-like intermediate, or (in a narrow range of crossover values of roundness) into a novel polycrystalline phase containing domains with square order in coexistence with clusters of particles having a weak hexagonal order.