Your assumptions (stereotypes) are quite wrong. Most HS grads couldn't even get in the military if they wanted to as 75% of young Americans have "issues" that prevent them from even being eligible to join. The average person in the military is smarter than the average American. 1 in 5 HS Grads can't pass the ASVAB (the test you take to see if you are smart enough to join the military and what kind of job to put you in if you do "pass"). They read at an entire grade level higher than people of the same age bracket. The average enlisted man comes from a better neighborhood too. You are right that they are more rural and southern, but they certainly ain't more "white"(75% of the US population is white, while only 73% in the military is). Blacks are slightly overrepresented in the military compared to rest of the nation, Hispanics underrepresented.
Overall, the make up of the enlisted military pretty much matches the US in general, but slightly smarter, slightly wealthier, slightly blacker, a little more rural, a little less urban, not as many poor, not as many rich, plenty of middle class.
Cites
here and
here.