They can't. EVER. The only thing they can "see" is what you tell them. There is no government database that they can use to access you medical history (so much red tape prevents this). It has been stated before and I'll reiterate... MEPS only knows what YOU tell them. Period. End of story.
TxIntel is right. Been to MEPS and I work in healthcare billing. There are privacy laws that prevent any medical provider from even talking to one another about a patient's medical condition except under only certain circumstances. For MEPS to access your medical history even with your consent would mean an insurmountable amount of work. I went through MEPS with 26 other recruits. They're not going to pay someone to do all that research. At the same time, if you have something chronic that's going to show up in training I'd do some hard thinking about it.