In this work, we study an unmanned aerial vehicle (UAV)-enabled secure integrated sensing and communication (ISAC) system, where a UAV serves as an aerial base station (BS) to simultaneously perform communication with a user and detect a target on the ground, while a dual-functional eavesdropper attempts to intercept the signals for both sensing and communication. Facing the dual eavesdropping threats, we aim to enhance the average achievable secrecy rate for the communication user by jointly designing the UAV trajectory together with the transmit information and sensing beamforming, while satisfying the requirements on sensing performance and sensing security, as well as the UAV power and flight constraints. To address the non-convex nature of the optimization problem, we employ the alternating optimization (AO) strategy, jointly with the successive convex approximation (SCA) and semidefinite relaxation (SDR) methods. Numerical results validate the proposed approach, demonstrating its ability to achieve a high secrecy rate while meeting the required sensing and security constraints.