Optical wireless communication (OWC) offers several complementary advantages to radio-frequency wireless networks such as its massive available spectrum; hence, it is widely anticipated that OWC will assume a pivotal role in the forthcoming sixth generation wireless communication networks. Although significant progress has been achieved in OWC over the past decades, the outage induced by occasionally low received optical power continues to pose a key limiting factor for its deployment. In this work, we discuss the potential role of single-photon counting (SPC) receivers as a promising solution to overcome this limitation. We present an overview of the applications of SPC-based OWC systems in 6G networks, introduce their major performance-limiting factors, propose a performance enhancement framework to tackle these issues, and identify critical areas of open problems for future research.