There is no way to predict what will happen if there is a shutdown. There is authority for the President to deliver "essential" services without an appropriation (shutdown). But, ATC is only essential if there are aircraft flying. So, ground all but military/emergency aircraft and the remaining ATC are not "essential".
Bottom line is that anybody who tells you what the President will do is either the President or doesn't know what they are talking about. If the former, please post what he tells you. If not, please don't.