Unmanned Aerial Vehicle (UAV) communications have recently entered a new period of interest, motivated by technological advances and the gradual emergence of the Space-Air-Ground Integrated Network (SAGIN). The current survey aims to capture the use of UAVs in the SAGIN while highlighting the most promising open research topics. The traditional UAV network architecture is not adequate to meet the challenges presented by the SAGIN, and an effective and secure space-air-ground integrated UAV network needs to be constructed. Given its well-distributed management and consensus mechanism, blockchain technology can make up for the deficiency of the traditional UAV network. In this work, we review the role of UAVs in the SAGIN. Then, three applications of the blockchain-envisioned UAV network are introduced through several classifications. Future challenges and the corresponding open research topics are also described.