Beam alignment (BA) is a crucial process in millimeter-wave (mmWave) communications for precise directional transmission and efficient link establishment, but its open nature makes it vulnerable to eavesdropping and identity impersonation attacks. To this end, we propose a novel security framework named CovertAuth, designed to provide a unified defense against both threats. For eavesdropping attacks, we advance the existing covert communication design by extending its joint optimization of the beam training budget and transmission power to a more practical BA scenario that incorporates the mutual coupling (MC) effect in antenna array impairments and imperfect eavesdropper channel state information. For impersonation attacks, the MC effect is explored as a device feature to design a novel adaptive weight-based physical layer authentication mechanism. Theoretical models for authentication metrics like detection and false alarm probabilities are first provided to conduct performance analysis. Based on these models, an optimization problem is constructed to determine the optimal weight value that maximizes authentication accuracy. With these optimal weights, a weighted-sum energy detector is employed to achieve identity validation. The resulting adaptive authentication scheme is then integrated with the covert communication to enhance the security of the BA phase. Finally, simulation results demonstrate that CovertAuth achieves high detection accuracy while satisfying the covertness requirement, offering a comprehensive security solution for the mmWave BA stage.