Fully integrated quantum technology based on photons is in the focus of current research, because of its immense potential concerning performance and scalability. Ideally, the single-photon sources, the processing units, and the photon detectors are all combined on a single chip. Impressive progress has been made for on-chip quantum circuits and on-chip single-photon detection. In contrast, nonclassical light is commonly coupled onto the photonic chip from the outside, because presently only few integrated single-photon sources exist. Here, we present waveguide-coupled single-photon emitters in the layered semiconductor gallium selenide as promising on-chip sources. GaSe crystals with a thickness below 100 nm are placed on Si3N4 rib or slot waveguides, resulting in a modified mode structure efficient for light coupling. Using optical excitation from within the Si3N4 waveguide, we find nonclassicality of generated photons routed on the photonic chip. Thus, our work provides an easy-to-implement and robust light source for integrated quantum technology.