A previously developed fundamental measure functional [J. Chem. Phys. 107, 6379 (1997)] is used to study the phase behavior of a system of parallel hard cubes. The single-component fluid exhibits a continuous transition to a solid with an anomalously large density of vacancies. The binary mixture has a demixing transition for edge-length ratios below 0.1. Freezing in this mixture reveals that at least the phase rich in large cubes always lies in the region where the uniform fluid is unstable, hence suggesting a fluid-solid phase separation. A method is developed to study very asymmetric binary mixtures by taking the limit of zero size ratio at fixed solvent fugacity. With this procedure the mixture is exactly mapped onto a one-component fluid of parallel adhesive hard cubes. At any density and solvent fugacity the large cubes are shown to collapse into a close-packed solid. Nevertheless the phase diagram contains a large metastability region with fluid and solid phases. Upon introduction of a slight polydispersity in the large cubes, the system shows the typical phase diagram of a fluid with an isostructural solid-solid transition (with the exception of a continuous freezing). Consequences about the phase behavior of binary mixtures of hard core particles are then drawn.