Many ultra-diffuse galaxies (UDGs) have been discovered in the Coma cluster, and there is evidence that some, notably Dragonfly 44, have Milky Way-like dynamical masses despite dwarf-like stellar masses. We used X-ray, UV, and optical data to investigate the star formation and nuclear activity in the Coma UDGs, and we obtained deep UV and X-ray data (Swift and XMM-Newton) for Dragonfly 44 to search for low-level star formation, hot circumgalactic gas, and the integrated emission from X-ray binaries. Among the Coma UDGs, we find UV luminosities consistent with quiescence but NUV$-r$ colors indicating star formation in the past Gyr. This indicates that the UDGs were recently quenched. The $r$-band luminosity declines with projected distance from the Coma core. The Dragonfly 44 UV luminosity is also consistent with quiescence, with SFR$<6times 10^{-4} M_{odot}$ yr$^{-1}$, and no X-rays are detected down to a sensitivity of $10^{38}$ erg s$^{-1}$. This rules out a hot corona with a $M > 10^8 M_{odot}$ within the virial radius, which would normally be expected for a dynamically massive galaxy. The absence of bright, low mass X-ray binaries is consistent with the expectation from the galaxy total stellar mass, but it is unlikely if most low-mass X-ray binaries form in globular clusters, as Dragonfly 44 has a very large population. Based on the UV and X-ray analysis, the Coma UDGs are consistent with quenched dwarf galaxies, although we cannot rule out a dynamically massive population.
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