We study the thermal transition between the paramagnetic and antiferromagnetic phases of interacting magnetic dipoles on square lattices in two dimensions. By Monte Carlo simulations, we find strong long-range order in the low temperature phase and a critical behavior that is, as in the fourfold anisotropic XY model, nonuniversal, in that the single-site quadrupolar anisotropy ${h}_{4}$ is a relevant variable. However, the critical indices differ markedly from the ones for the anisotropic XY model. They vary from Ising type, within errors, for ${h}_{4}=0$, to a critical behavior in the ${h}_{4}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit that is nearly as abrupt as a first-order transition.