Spin squeezing is a genuine multipartite entanglement resource enabling quantum-enhanced metrology beyond classical limits. Based on a hybrid solid-state system of nitrogen-vacancy (NV) centers and magnons, we propose a novel scheme for generating spin squeezing via Kerr nonlinearity. In the large-detuning regime, by tuning the strength of the magnon parametric amplification (MPA) enabled by Kerr effect, the magnon-mediated spin-spin interactions can be controlled, thereby allowing flexible switching among one-axis twisting (OAT), two-axis twisting (TAT) and unconventional two-axis twisting (UTAT). Remarkably, in the presence of realistic dissipation and finite-size effects, an optimized UTAT interaction exhibits enhanced spin squeezing and quantum-enhanced sensing performance compared with the corresponding OAT and TAT cases. Furthermore, in the resonant and strongly dissipative regime, the MPA-induced TAT interaction and squeezed magnon reservoir jointly enable steady-state spin squeezing of the NV ensemble. Our results establish a unified and flexible scheme for generating both dynamical and steady-state spin squeezing, with potential applications in quantum-enhanced precision measurements.