I don't think so. The constitution is a limitation on the powers of government, not on the actions of citizens. If the airlines or other NGO were to take the security of their property seriously (property = airplanes, flight crews, revenue paying passengers) and be wholly, and comprehensively responsible for them in exchange for the fare and conditions of carriage, they could require real security screening in addition to the TSA. They don't because they're indemnified as of 9/11/01.

The states may enact laws to protect its citizens from arbitrary discrimination and most states have public accommodation laws on the books which invoke criminal and tort liability for arbitrary discrimination, but I don't think there is anything that would prohibit an airline from setting up its own security screening (in addition to the TSA screening) and asking you if you what ever it wants, consistent with state accommodation laws.

I'm not sure that if a theoretical M-Go Blue airlines set up such a supplemental screening and ask its passengers if they liked Buckeyes and denied boarding based on the answer would meet the test of Michigan non-discrimination test, though.