In recent years, unmanned aerial vehicles (UAVs) have become a key role in wireless communication networks due to their flexibility and dynamic adaptability. However, the openness of UAV-based communications leads to security and privacy concerns in wireless transmissions. This paper investigates a framework of UAV covert communications which introduces flexible reconfigurable intelligent surfaces (F-RIS) in UAV networks. Unlike traditional RIS, F-RIS provides advanced deployment flexibility by conforming to curved surfaces and dynamically reconfiguring its electromagnetic properties to enhance the covert communication performance. We establish an electromagnetic model for F-RIS and further develop a fitted model that describes the relationship between F-RIS reflection amplitude, reflection phase, and incident angle. To maximize the covert transmission rate among UAVs while meeting the covert constraint and public transmission constraint, we introduce a strategy of jointly optimizing UAV trajectories, F-RIS reflection vectors, F-RIS incident angles, and non-orthogonal multiple access (NOMA) power allocation. Considering this is a complicated non-convex optimization problem, we propose a deep reinforcement learning (DRL) algorithm-based optimization solution. Simulation results demonstrate that our proposed framework and optimization method significantly outperform traditional benchmarks, and highlight the advantages of F-RIS in enhancing covert communication performance within UAV networks.