Hmm. Many many things would happen indeed. Just one? Okay. Countless ships at sea would either fall off course or crash onto land. This would in turn destroy the economy, or at least leave it quite ravaged. I thought perhaps they could set a 24 hour auto-pilot course to avoid this, but you said they skip a day, not plan time off. So every person on those ships would just say "Screw it, somebody else will cover it" but nobody does, because they too are skipping work.
Now, would interfering with a large scale disaster count as work, or simple survival? If I was on an oil tanker and we were about to get perpendicular with a sharp, rocky coastline, I wouldn't not get behind the wheel and turn the boat 90deg, because its my day off, damn it, I'm going to live to see all of it, not die for the sake of some philosophical proposition. Or is this a sort of bystander effect, everyone waiting to see what someone else will do to save them all, as the though of doing it themselves does not cross their minds as per the parameters of your hypothetical?
OR, would they just get the cook to do it? Its not his job, so he wouldn't be at work, but it is a job, so he would be working. I feel this is my impasse via infinite loop.
As for the bonus: Does writing count? I would assume yes, so none of that. Well. I don't know about the rest of the world or what ruin may befall it, but I will go hiking. I prefer not to talk and just soak up the trees 'n shit, and don't have a cellphone, so just being alone in nature would allow me to feel as if no change had ever taken place. That's what would happen in this very small neck of the woods, anyway.
"I'm not stupid, I just have a command of thoroughly useless information." Watterson