We propose that there exists a class of transient sources, "squeezars", which are stars caught in highly eccentric orbits around a massive (m<108 Mo) black hole (MBH), whose atypically high luminosity (up to a significant fraction of their Eddington luminosity) is powered by tidal interactions with the MBH. Their existence follows from the presence of a mass sink, the MBH, in the galactic center, which drives a flow of stars into nearly radial orbits to replace those it has destroyed. We consider two limits for the stellar response to tidal heating: surface heating with radiative cooling ("hot squeezars") and bulk heating with adiabatic expansion ("cold squeezars"), and calculate the evolution of the squeezar orbit, size, luminosity and effective temperature. The squeezar formation rate is only 0.05 that of tidal disruption flares, but squeezar lifetimes are many orders of magnitude longer, and so future observations of squeezars in nearby galaxies can probe the tidal process that feeds MBHs and the effects of extreme tides on stars. The mean number of squeezars orbiting the Galactic MBH is estimated at 0.1-1.