This paper proposes a secure downlink multi-user transmission scheme enabled by a flexible unmanned aerial vehicle base station (UAV-BS) and non-orthogonal multiple access (NOMA). According to their heterogeneous service requirements, multiple legitimate users are categorized as security-required users (SUs) and quality of service (QoS)-required users (QUs), while these QUs can potentially act as internal eavesdroppers which are curious about the secrecy transmissions of SUs. In such a context, our goal is to maximize the achievable minimum secrecy rate among SUs through the joint optimization of user scheduling, power allocation, and trajectory design, subject to the QoS requirements of QUs and the mobility constraint of UAV-BS. Due to the non-convexity of the problem, an efficient iterative algorithm is firstly proposed, based on the alternative optimization (AO) and successive convex approximation (SCA) methods and along with a penalty-based algorithm to deal with the introduced binary integer variables, to obtain a sub-optimal solution. Then, we propose an SUs-oriented low-complexity algorithm by taking advantage of the inherent characteristics of the optimization problem, which can efficiently reduce the computational complexity and can act as a reasonable initial solution for the previous iterative algorithm to achieve better performance. Finally, the superiority of our proposed scheme compared with the conventional orthogonal multiple access (OMA) one is validated by numerical simulation results.