Safety-critical and mission-oriented applications have seen an increase in the use of Unmanned Aerial Vehicles (UAVs). These require reliable real-time flight control and robust security guarantees. Modern UAV platforms are integrated with sensing, computation, communication, and control capabilities utilizing embedded System-on-Chip (SoC) architectures. However, with increasing levels of autonomy, connectivity, and computational complexity, UAVs are all exposed to many types of cyber-physical security threats, both in dual-use and the civilian sector. Examples of threat actors to a UAV include communication spoofing, sensor manipulation, firmware tampering, and hardware Trojan attacks. This paper is a comprehensive survey of the state-of-the-art secure and reconfigurable System-on-Chips (SoCs) and SoC architectures for UAV flight control applications. It provides a systematic review of UAV flight control system architectures, embedded computing platforms, and UAV cyber-physical system-specific security challenges. Additionally, it defines a UAV flight control-specific threat model and attack taxonomy and discusses vulnerabilities associated with the communication, sensing, control, software, and hardware levels of a UAV. It also discusses the role of FPGA-based and FPGA-SoC platforms as a means of augmenting hardware-assisted security and runtime adaptability. Finally, the authors identify key research gaps and propose control-aware, dynamically reconfigurable secure SoC architectures for the next generation of UAV flight controllers.