On terrestrial planets with thin atmospheres, the abundances of key trace gases such as methane (CH4) are controlled by photochemistry and source fluxes such as the rate of volcanic outgassing, water-rock reactions, or biological production. The interpretation of methane as a biosignature is thus ultimately dependent on the production flux inferred from its abundance and the likelihood that this flux could be produced by geological sources alone. Prior work has shown that the buildup of methane in the atmosphere at a given flux is highly favored for planets with oxygen-rich atmospheres orbiting M dwarfs, relative to Sun-like stars, due to reduced NUV photons that drive key photochemical destruction pathways. However, relatively limited attention has been given to anoxic, Archean-like atmospheres and their flux-abundance relationships. We use a photochemical model to predict the atmospheric CH4 mixing ratio as a function of its production rate for planets in the habitable zones of FGKM stars. We then compare the fluxes to those produced by primitive bacterial biospheres and geologic processes to evaluate what levels of CH4 would suggest biological activity. We find that the shape of the flux-abundance relationship is highly dependent on the host star spectrum, as are the photochemical pathways for CH4 destruction. For example, at low CH4 fluxes/abundances, CH4 destruction by the OH radical dominates for planets orbiting all stellar types but is much less efficient for planets orbiting M dwarfs, which may challenge biosignature interpretations. At high CH4 fluxes/abundances, CH4 photolysis is dominant for GKM stars, but not F stars where destruction by OH is always the dominant path. In general, the shape and magnitude of the stellar UV spectrum is predictive of these relationships.