In this paper, a new methodology is developed for modeling and analyzing a full-duplex UAV-assisted relay system to facilitate solving the problems of UAV energy constraints and the vulnerability of UAVs to eavesdropping in the air. Combining simultaneous wireless information and power transfer (SWIPT) technology, we model the downlink UAV eavesdropping channel and propose a secure transmission protocol for a full-duplex UAV-assisted relay system. Using this transmission protocol, we analyze and derive the connectivity and security of the entire communication link, including connection probability and lower bounds on secrecy outage probability. A key intermediate step in our analysis is to derive the signal-to-digital noise ratio of the destination and eavesdropping nodes of the full-duplex UAV relay link. The analyses show that the power allocation factor is a trade-off between system connectivity and security, while greater eavesdropping interference needs to be sacrificed for an equal magnitude of security performance improvement under high security demand conditions.