Those saying they've done this and have had it go well is because the company is thrilled that you'll fly sick and they've been lucky to not have anything bad happen on these flights and have things "looked into."

If you are legitimately sick, don't worry about catching flack, the heat being put on you, etc for calling in sick, even if it is one minute before departure. It isn't safe, it isn't legal, and people get sick... calling in sick is a perfectly normal thing to do.