We analyse the structure of galaxies with high specific star formation rate (SSFR) in cluster and field environments in the redshift range \(0.5<z<1.0\). Recent studies have shown that these galaxies are strongly depleted in dense environments due to rapid environmental quenching, giving rise to post-starburst galaxies (PSBs). We use effective radii and S\'ersic indices as tracers of galaxy structure, determined using imaging from the UKIDSS Ultra Deep Survey (UDS). We find that the high-SSFR galaxies that survive into the cluster environment have, on average, larger effective radii than those in the field. We suggest that this trend is likely to be driven by the most compact star-forming galaxies being preferentially quenched in dense environments. We also show that the PSBs in clusters have stellar masses and effective radii that are similar to the missing compact star-forming population, suggesting that these PSBs are the result of size-dependent quenching. We propose that both strong stellar feedback and the stripping of the extended halo act together to preferentially and rapidly quench the compact and low-mass star-forming systems in clusters to produce PSBs. We test this scenario using the stacked spectra of 124 high-SSFR galaxies, showing that more compact galaxies are more likely to host outflows. We conclude that a combination of environmental and secular processes is the most likely explanation for the appearance of PSBs in galaxy clusters.