We present an analysis of a sample of flares on ``quiescent'' (i.e. non-flare) M and K stars using temporally resolved UV spectroscopy from the growing body of MUSCLES Treasury Survey data. Specifically, our analysis quantified the response of the far-UV C II, Si III, Si IV, and N V emission lines and the far-UV continuum during the flares. Using these tracers, we examined one representative event on GJ 832. In concordance with flares recorded on the Sun and AD Leo, the MUSCLES flares are well fit by a power law relationship of similar slope in frequency versus energy. Flares can strip atmospheric mass from orbiting planets, adversely affecting their long-term habitability. To gauge the amplitude of this effect, we computed an energy-balance upper-limit on the amount of atmosphere a large flare might remove from an orbiting Earth due purely to elevated EUV flux and found this limit to be modest relative to Earth's atmospheric mass.