This paper introduces a novel approach to enhance the secrecy performance of nonorthogonal multiple access (NOMA) systems by leveraging multiple unmanned aerial vehicles (UAVs) equipped with intelligent reflecting surfaces (IRSs). In this proposed system, multiple UAVs mounted with IRSs (shortened as U/Ss) are strategically deployed to assist two legitimate users in the presence of multiple colluding eavesdroppers (CEs) or noncolluding eavesdroppers (NCEs) attempting to intercept messages. The paths from the transmitter to the users are combined with those involving the U/Ss to maximize the received message power. We derive mathematical expressions for the secrecy capacities (SCs) of the proposed U/S-NOMA systems over Nakagami- channels. Additionally, asymptotic expressions for SCs in the high transmit power region are provided. Numerical results demonstrate that the SCs of U/S-NOMA systems significantly surpass those of traditional NOMA networks lacking U/Ss. Notably, the U/S-NOMA systems achieve their highest SCs more rapidly than traditional NOMA systems. Consequently, the integration of U/Ss proves effective in reducing transmit power and enhancing the secrecy performance of NOMA systems. Furthermore, we also delve into the impact of key parameters such as the number of reflecting elements (REs) in U/Ss, carrier frequency, U/Ss' positions, fading order, bandwidth, number of eavesdroppers, and NOMA power allocation coefficients. Valuable recommendations are presented based on a thorough investigation of these crucial parameters.