We present structural measurements of 145 spectroscopically selected intermediate-redshift (z~0.7), massive (M_*_~10^11^M_{sun}_) post-starburst galaxies from the SQuIGG->LE sample measured using wide-depth Hyper Suprime-Cam i-band imaging. This deep imaging allows us to probe the sizes and structures of these galaxies, which we compare to a control sample of star-forming and quiescent galaxies drawn from the LEGA-C Survey. We find that post-starburst galaxies systematically lie ~0.1dex below the quiescent mass-size (half-light radius) relation, with a scatter of ~0.2dex. This finding is bolstered by nonparametric measures, such as the Gini coefficient and the concentration, which also reveal these galaxies to have more compact light profiles than both quiescent and star-forming populations at similar mass and redshift. The sizes of post-starburst galaxies show either negative or no correlation with the time since quenching, such that more recently quenched galaxies are larger or similarly sized. This empirical finding disfavors the formation of post-starburst galaxies via a purely central burst of star formation that simultaneously shrinks the galaxy and shuts off star formation. We show that the central densities of post-starburst and quiescent galaxies at this epoch are very similar, in contrast with their effective radii. The structural properties of z~0.7 post-starburst galaxies match those of quiescent galaxies that formed in the early universe, suggesting that rapid quenching in the present epoch is driven by a similar mechanism to the one at high redshift.