The model system of parallel hard cubes is studied by using fundamental measure theory (FMT) and extensive Monte Carlo simulations. A continuous freezing transition occurs in this system to which finite-size scaling analysis is applied. Significant deviations from a previous simulation study are found for the position of the critical point and for the critical exponents. Our results are compatible with the Heisenberg universality class. Moreover, both theory and simulation show that also at high densities the solid phase is thermodynamically more stable than a possible columnar phase. FMT appears quantitatively more reliable at high densities than near the critical density, which is substantially underestimated.