10 April Fool’s Day.
Why it should be a public holiday: Every year, people inevitably get fired for ill-advised April Fool’s pranks at work, so we’d trade this one day off for those people not winding up with unlimited days off.
Why it should not be a public holiday: Yeah, this is the least likely candidate on the list, but at least it made the list, unlike Earth Day, Groundhog Day and Talk Like a Pirate Day.