The reconfigurable intelligent surface (RIS) technology shows great potential in sixth-generation (6G) terrestrial and non-terrestrial networks (NTNs) since it can effectively change wireless settings to improve connectivity. Extensive research has been conducted on traditional RIS systems with diagonal phase response matrices. The straightforward RIS architecture, while cost-effective, has restricted capabilities in manipulating the wireless channels. The beyond diagonal reconfigurable intelligent surface (BD-RIS) greatly improves control over the wireless environment by utilizing interconnected phase response elements. This work proposes the integration of unmanned aerial vehicle (UAV) communications and BD-RIS in 6G NTNs, which has the potential to further enhance wireless coverage and spectral efficiency. We begin with the preliminaries of UAV communications and then discuss the fundamentals of BD-RIS technology. Subsequently, we discuss the potential of BD-RIS and UAV communications integration. We then proposed a case study based on UAV-mounted transmissive BD-RIS communication. Finally, we highlight future research directions and conclude this work.