We investigate square-lattice skyrmion crystal states in a centrosymmetric spin model with momentum-dependent magnetic interactions, including a uniform ferromagnetic interaction, easy-axis anisotropic interactions at two orthogonal finite wave vectors, higher-harmonic wave-vector interactions, and an external magnetic field. Using simulated annealing calculations, we determine the low-temperature magnetic phase diagram as functions of magnetic field and uniform ferromagnetic interaction strength. We find that the skyrmion crystal phase is stabilized at intermediate fields when the uniform ferromagnetic interaction is weak, exhibiting a noncoplanar double-Q spin texture, finite scalar spin chirality, and higher-harmonic components. Increasing the uniform ferromagnetic interaction suppresses and eventually eliminates the skyrmion crystal phase through direct first-order transitions to the fully polarized state, whereas its weakening expands the skyrmion stability region and stabilizes a single-Q conical spiral phase at high fields. These results clarify the mechanisms governing the square-lattice skyrmion formation in centrosymmetric magnets.