<p>We study theoretically the achievement of strong mechanical squeezing and entanglement in a magnomechanical system consisting of a magnon mode and a mechanical mode, where the magnon mode is driven by a periodically modulated microwave field. By utilizing the self-cooling mechanism inherent in the cavity magnomechanics and combining it with the nonlinear effects induced by the periodic amplitude modulation, the steady-state mechanical squeezing and bipartite entanglement can be generated not only in the resolved sideband regime but also in the unresolved sideband regime. Interestingly, influenced by the periodic modulation, strong mechanical squeezing can be achieved in the resolved sideband regime. Moreover, both the mechanical squeezing and the entanglement exhibit strong robustness against the environmental temperature.</p>