This paper investigates the potential of unmanned aerial vehicles (UAVs) equipped with movable-antenna (MA) arrays to strengthen security in wireless communication systems. We propose a novel framework that jointly optimizes the UAV trajectory and the reconfigurable beamforming of the MA array to maximize secrecy energy efficiency, while ensuring reliable communication with legitimate users. By exploiting the spatial degrees of freedom enabled by the MA array, the system can form highly directional beams and deep nulls, thereby significantly improving physical layer security. Numerical results demonstrate that the proposed approach achieves superior secrecy energy efficiency, attributed to the enhanced spatial flexibility provided by the movable antenna architecture.