In this study, we investigate a hybrid system consisting of an atomic ensemble trapped inside a dissipative optomechanical cavity assisted with perturbative oscillator-qubit coupling. Such a system is generally very suitable for generating stationary squeezing of the mirror motion in the long-time limit under the unresolved sideband regime. Based on the master equation and covariance matrix approaches, we discuss in detail the respective squeezing effects. We also determine that in both approaches, simplifying the system dynamics with adiabatic elimination of the highly dissipative cavity mode is very effective. In the master equation approach, we find that the squeezing is a resulting effect of the cooling process and is robust against thermal fluctuations of the mechanical mode. In the covariance matrix approach, we can approximately obtain the analytical result of the steady-state mechanical position variance from the reduced dynamical equation. Finally, we compare the two approaches and observe that they are completely equivalent for the stationary dynamics. Moreover, the scheme may be useful for possible ultraprecise quantum measurement that involves mechanical squeezing.