This paper investigates the utilization of simultaneously transmitting and reflecting RIS (STAR-RIS) in supporting joint physical layer security (PLS) and covert communications (CCs) in a multi-antenna millimeter wave (mmWave) system, where the base station (BS) communicates with both covert and security users while defeating eavesdropping by wardens with the help of a STAR-RIS. Specifically, analytical derivations are performed to obtain the closed-form expression of warden's minimum detection error probability (DEP). Furthermore, the asymptotic result of the minimum DEP and the lower bound of the secure rates are derived, considering the practical assumption that BS only knows the statistical channel state information (CSI) between STAR-RIS and the wardens. Subsequently, an optimization problem is formulated with the aim of maximizing the average sum of the covert rate and the minimum secure rate while ensuring the covert requirement and quality of service (QoS) for legal users by jointly optimizing the active and passive beamformers. Due to the strong coupling among variables, an iterative algorithm based on the alternating strategy and the semi-definite relaxation (SDR) method is proposed to solve the non-convex optimization problem. Simulation results indicate that the performance of the proposed STAR-RIS-assisted scheme greatly surpasses that of the conventional RIS scheme, which validates the superiority of STAR-RIS in simultaneously implementing PLS and CCs.