  (Abridged) Star formation-driven outflows are a critical phenomenon in
theoretical treatments of galaxy evolution, despite the limited ability of
observations to trace them across cosmological timescales. If the strongest
MgII absorption-line systems detected in the spectra of background quasars
arise in such outflows, "ultra-strong" MgII (USMgII) absorbers would identify
significant numbers of galactic winds over a huge baseline in cosmic time, in a
manner independent of the luminous properties of the galaxy. To this end, we
present the first detailed imaging and spectroscopic study of the fields of two
USMgII absorber systems culled from a statistical absorber catalog, with the
goal of understanding the physical processes leading to the large velocity
spreads that define such systems. Each field contains two bright emission-line
galaxies at similar redshift (dv < 300 km/s) to that of the absorption.
Lower-limits on their instantaneous star formation rates (SFR) from the
observed OII and Hb line fluxes, and stellar masses from spectral template
fitting indicate specific SFRs among the highest for their masses at z~0.7.
Additionally, their 4000A break and Balmer absorption strengths imply they have
undergone recent (~0.01 - 1 Gyr) starbursts. The concomitant presence of two
rare phenomena - starbursts and USMgII absorbers - strongly implies a causal
connection. We consider these data and USMgII absorbers in general in the
context of various popular models, and conclude that galactic outflows are
generally necessary to account for the velocity extent of the absorption. We
favour starburst driven outflows over tidally-stripped gas from a major
interaction which triggered the starburst as the energy source for the majority
of systems. Finally, we discuss the implications of these results and speculate
on the overall contribution of such systems to the global SFR density at z~0.7.
