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We demonstrate how magnon Kerr nonlinearity creates bistability of quantum states when the magnon mode is strongly driven by a microwave field in a nonlinear cavity magnonic system. Numerical simulation results with experimentally feasible parameters show that the method where the system is driven far from equilibrium is a reliable way to achieve squeezed states for magnons and photons and to carry out magnon-photon entanglement, revealing mysterious phenomena of bistability. In addition, the Kittel mode can jump from one state to another one near two switching points, thus achieving the hysteresis loop phenomenon. Our results indicate that bistable quantum states could provide a way to study macroscopic quantum bistable phenomena in nonlinear systems and also can be found in broad applications in magnetic spintronics.