Quenched galaxies at z > 2 are nearly all very compact relative to z similar to 0, suggesting a physical connection between high stellar density and efficient, rapid cessation of star-formation. We present rest-frame UV spectra of Lyman-break galaxies (LBGs) at z similar to 3 selected to be candidate progenitors of the quenched galaxies at z similar to 2 based on their compact rest-frame-optical sizes and high Sigma(SFR). We compare their UV properties to those of more extended LBGs of similar mass and star-formation rate (non-candidates). We find that candidate progenitors have faster bulk interstellar medium (ISM) gas velocities and higher equivalent widths of interstellar absorption lines, implying larger velocity spread among absorbing clouds. Candidates deviate from the relationship between equivalent widths of Ly alpha and interstellar absorption lines in that their Ly alpha emission remains strong despite high interstellar absorption, possibly indicating that the neutral H I fraction is patchy, such that Ly alpha photons can escape. We detect stronger CIV P-Cygni features (emission and absorption) and He II emission in candidates, indicative of larger populations of metal-rich Wolf-Rayet stars compared to non-candidates. The faster bulk motions, broader spread of gas velocity, and Ly alpha properties of candidates are consistent with their ISM being subject to more energetic feedback than non-candidates. Together with their larger metallicity (implying more evolved star-formation activity) this leads us to propose, if speculatively, that they are likely to quench sooner than non-candidates, supporting the validity of selection criteria used to identify them as progenitors of z similar to 2 passive galaxies. We propose that massive, compact galaxies undergo more rapid growth of their stellar mass content, perhaps because the gas accretion mechanisms are different, and quench sooner than normally sized LBGs at these (early) epochs.