There are no rules. Players get to give a shout out to one school and they say whatever they want. Would the networks let a player who went to Miami say they went to Oregon? Probably not. But as long as a player played at a school they can mention that school.
Some players mention their high school out of pride, others have done it because they left on bad terms with their college and don't want to mention them, and some go back and forth saying their high school sometimes and their college sometimes.
If a player says which high school they went to I wouldn't read into that at all since 99%+ attended some college to play football at some point - it might not be DI and could even be community college but there are very very players in the NFL that didn't play in college (I know a few foreign born kickers and a couple rugby players).
The other note is that players have pretty much a free reign on how they say the name of their college. The word "THE" has been added to the name of many colleges to I guess, gain more respect. The whole naming of your school is just a fun thing that a player gets to do for each game. These are usually done the day before and fresh for NFL games. They also need to get 11 guys to say their college in between a play - about 40 seconds. So it is quite funny watching guys try to say the name of their college as slow as possible.