The specific questions which I've seen asked by airlines all relate to drug tests administered under a formal DOT drug test program (ie truck drivers, airline pilots, etc). IIRC it only goes back a couple years, I think there are legal limits on what employers can ask.
However...airlines can and do ask about your military record. So a very critical point here is did you actually begin military service? If so you would have received a DD-214 with a discharge code which may indicate lack of suitability or possibly even specific drug test failure.
If you just started the MEPS process, then you would probably not have actually entered military service and may not have to address this issue with airlines. But don't lie to the FAA, read the medical form carefully.
I doubt there is any way for an employer background check to find out about MEPS.