Superconducting nanowire single photon detectors (SNSPDs) made with thin NbN films can reach high performances. While sputtering has been the deposition method of choice, here, we show that ammonia-molecular beam epitaxy (NH3-MBE) can produce pertinent epitaxial cubic NbN thin films on silicon substrates using an AlN buffer. Despite granular morphology and a high density of grain boundaries as well as the presence of rotational twins, Tc = 12.7 K for a 5.6 nm thick film and saturation of internal detection efficiency up to 850 nm are achieved. Morphology and stoichiometry as well as strain have a strong impact on the detector properties, highlighting the importance of a precise control of the growth parameters. These results pave the way for high fabrication yield of SNSPDs on large-scale silicon wafers using epitaxial NbN thin films grown by MBE.