UAVs are increasingly appearing in swarms or formations to leverage cooperative behavior, forming flying ad hoc networks. These UAV-enabled networks can meet several complex mission requirements and are seen as a potential enabler for many of the emerging use-cases in future communication networks. Such networks, however, are characterized by a highly dynamic and mobile environment with no guarantee of a central network infrastructure which can cause both connectivity and security issues. While wireless mesh networks are envisioned as a solution for such scenarios, these networks come with their own challenges and security vulnerabilities. In this paper, we analyze the key security and resilience issues resulting from the application of wireless mesh networks within UAV swarms. Specifically, we highlight the main challenges of applying current mesh technologies within the domain of UAV swarms and expose existing vulnerabilities across the communication stack. Based on this analysis, we present a security-focused architecture for UAV mesh communications. Finally, from the identification of these vulnerabilities, we discuss research opportunities posed by the unique challenges of UAV swarm connectivity.