They're not going to kick you out because you were legitimately sick. If you call in on a holiday, they typically make you get a doctor's note anyway. I think the issue comes in more when you get asked to provide a doctor's note and then can't provide it, which under the current CBA would be considered UNAV, which would probably disqualify you anyway.
And even if they did disqualify people for being sick, I wouldn't want to fly for a company like that anyway.