A flux of neutrons from an astrophysical source in the Galaxy can be detected in the Pierre Auger Observatory as an excess of cosmic-ray air showers arriving from the direction of the source. To avoid the statistical penalty for making many trials, classes of objects are tested in combinations as nine 'target sets', in addition to the search for a neutron flux from the Galactic Center or from the Galactic Plane. Within a target set, each candidate source is weighted in proportion to its electromagnetic flux, its exposure to the Auger Observatory, and its flux attenuation factor due to neutron decay. These searches do not find evidence for a neutron flux from any class of candidate sources. Tabulated results give the combined p-value for each class, with and without the weights, and also the flux upper limit for the most significant candidate source within each class. These limits on fluxes of neutrons significantly constrain models of EeV proton emission from non-transient discrete sources in the Galaxy.