Unmanned aerial vehicles (UAVs) have attracted a lot of research attention because of their high mobility and low cost in serving as temporary aerial base stations (BSs) and providing high data rates for next-generation communication networks. To protect user privacy while avoiding detection by a warden, we investigate a jammer-aided UAV covert communication system, which aims to maximize the user's covert rate with optimized transmit and jamming power. The UAV is equipped with multi-antennas to serve multi-users simultaneously and enhance the Quality of Service. By considering the general composite fading and shadowing channel models, we derive the exact probability density (PDF) and cumulative distribution functions (CDF) of the signal-to-interference-plusnoise ratio (SINR). The obtained PDF and CDF are used to derive the closed-form expressions for detection error probability and covert rate. Furthermore, the covert rate maximization problem is formulated as a Nash bargaining game, and the Nash bargaining solution (NBS) is introduced to investigate the negotiation among users. To solve the NBS, we propose two algorithms, i.e., particle swarm optimization-based and joint twostage power allocation algorithms, to achieve covertness and high data rates under the warden's optimal detection threshold. All formulated problems are proven to be convex, and the complexity is analyzed. The numerical results are presented to verify the theoretical performance analysis and show the effectiveness and success of achieving the covert communication of our algorithms.