The NAAT test at Walgreens is a PCR test, not antigen like CVS or the majority of home tests. From what I've read, you may pop a positive with PCR/NAAT if you had covid in the recent past (can't define recent, I've seen people say 30-90 days). I believe is also a more accurate test with less likelihood for false negatives than an antigen test but I could be wrong.
In any case, my point is there was likely a scientific reason that this person tested positive with a NAAT test but not with an antigen test and it has nothing to do with Walgreens.