The effect of an external perpendicular magnetic field on the magnetization and crystallization of magnetic particles trapped in multilayered arrays is investigated theoretically. General ground-state symmetry considerations corroborated by numerical results allow us to develop an exact analytical theory. When dealing with monolayers, the continuous degeneracy of the in-plane dipole orientation is broken and accompanied by a rhombic crystal structure at finite applied magnetic field. On the other hand, bilayer systems are more robust against applied magnetic fields and preserve the in-plane hexagonal lattice symmetry. The onset of disaggregation is predicted as well in both scenarios. Our findings can be experimentally verified in colloidal suspensions (typically ferrofluids or magnetorheological fluids) and provide enlightening hints to further understanding the phase behavior of magnetic multilayers in more conventional molecular and atomic materials such as thin metal films obtained by epitaxy in the presence of an external magnetic field.