I'm more and more convinced that Jax's "thing" that happens when he holds his breath is that his pupils widen. That would explain why he doesn't want to tell the others, because unless he's caught off guard he does it constantly. We know the characters don't actually need to breath but it still feels like choking to them. Jax is trying so hard to convince himself that he is just a cartoon character (and maybe there's also a part of self-inflicted punishment in it because he feels guilty about not being able to save Ribbit) that he's basically torturing himself every second.
The "shocking thing that will recontextualize a character and piss off a specific part of the fanbase" Goose told about will be this reveal. Imagine all the drawfags realizing that all their cute pictures of their rabbit husbando have retroactively been changed to have a self-harming subtext.