We theoretically propose a scheme based on the Barnett effect to investigate the nonreciprocal quantum squeezing of the magnon and cavity modes in a cavity magnonic system. The proposed setup considers a typical cavity magnonic system in which a yttrium iron garnet (YIG) sphere is placed inside a highquality microwave cavity. Under an external static magnetic field, the YIG sphere supports collective spin excitations, namely the magnon mode, which can strongly couple to the cavity mode via magnetic dipole interaction. The system can be described by the standard cavity–magnon Hamiltonian, which includes the free-energy terms of the cavity and magnon modes as well as their interaction term. In this model, a squeezed driving field is further injected into the microwave cavity. By employing the Heisenberg–Langevin equations, the dynamical equations of the system operators can be derived. The system operators are then decomposed into their steady-state mean values and quantum fluctuation parts. By performing linearization around the steady state, one obtains the linearized quantum Langevin equations that describe the dynamics of quantum fluctuations. By rewriting these equations in a matrix form and solving the corresponding covariance matrix, the squeezing properties of the cavity and magnon modes and their dependence on system parameters can be quantitatively analyzed. To introduce nonreciprocal effects, we further consider the rotation of the YIG sphere. According to the Barnett effect, the spin system experiences an effective magnetic field when the magnetic medium rotates, which leads to an additional frequency shift in the magnon mode that depends on the rotational angular velocity, known as the Barnett frequency shift. The sign of this shift depends on both the rotation direction and the orientation of the external bias magnetic field. Therefore, by reversing the direction of the bias magnetic field, either a positive or negative Barnett frequency shift can be realized. This mechanism breaks the symmetry of the system under different magnetic field directions, resulting in pronounced nonreciprocal squeezing behavior. Specifically, squeezing only appears for one particular magnetic field orientation, while it is significantly suppressed or even completely absent for the opposite orientation. Moreover, the squeezing of both the cavity and magnon modes exhibits strong robustness against thermal effects. The nonreciprocal squeezing can still be maintained at temperatures of about 1 K, which relaxes the stringent requirements on experimental conditions. Our work provides a new physical mechanism for realizing nonreciprocal quantum state control in cavity magnonic systems and may have potential applications in quantum information processing and the integration of chiral quantum devices.