By comparing solar data with data of ``flat activity'' stars at UV and soft X-ray wavelengths, we estimate solar chromospheric, transition region, and coronal emission as it might have existed during the Maunder minimum (1645-1715), the most recent of several episodes of persistently weak sunspot activity. Several flat activity stars from the Mount Wilson S-index survey have been observed with the Hubble Space Telescope and ROSAT. Drawing on published data, we reassess the criteria by which a star may be considered as a Maunder minimum candidate. Of our targets, HD 10700 and HD 143761 are the most likely such candidates. Solar UV spectra from the SOHO and UARS spacecraft, and soft X-ray data from the SNOE spacecraft are compared with the stellar UV and X-ray data. The comparison suggests that the radiative output of the Maunder minimum chromosphere, transition region, and corona were similar to (or at least not much less than) those observed under conditions close to current solar minima. In turn, this suggests that the emitting structures (which on the Sun at sunspot minimum are small scale, <<Rsolar), including the magnetic network, were also similar. These results have implications for the nature of the surface magnetic fields and irradiance during the Maunder and other magnetic minima, and for the solar dynamo.