IMO, there should be some indication of what exactly makes them sick, it almost seems random. I know some foods seem to really make them sick easily, but I'm not really sure why they get sick otherwise. Seems like maybe it's a chance based on how big a certain food is compared to your stomach capacity, IDK.
Either way, I skipped the dodgy mechanic entirely by mass-purchasing cheap, small (and small-ish) foods, then stockpiling certain badges to make the small foods more useful, then I use an autoclicker on my mouse to eat thousands of them individually, my character never gets sick anymore, and I get to chuckle at the mints' description about getting immobile from mints.