Colloidal particles with anisotropic shapes and interactions display rich phase behavior and have potential as structural bases for materials with controllable properties. In this paper, we explore the self-assembling characteristics of a new class of particles that have been shown experimentally to form reconfigurable structures: microscopic cube-shaped colloids with a magnetic dipole that is transversely offset from the particle's center of mass. We have performed in silico simulations of the dynamics of large numbers of dipolar squares in two-dimensions using discontinuous molecular dynamics (DMD). We use a coarse-grain method where the dipolar microcubes are represented by a group of four hard circles bonded together to create a rigid square in two-dimensions and two opposite charges are embedded within the square to represent a magnetic dipole. Annealing, or "slow-cooling", simulations are conducted to determine the equilibrium structures. Systems of dipolar squares tend to assemble into one of two different types of conformations: either single- or double-stranded assemblies, each with unique structures and phase diagrams in the temperature-density plane. Single-stranded assemblies form highly interconnected percolated, or gel-like, networks. In contrast, double stranded assemblies tend to form globally-aligned nematic states at high densities, although this is not seen consistently in all runs. The phase behavior of systems of dipolar squares depends not only on the system's temperature and density, but also on the type of dipole embedded within the square and on the relative number of squares with an opposite "handedness" that are present within the system.