It looks like homeschooled kids are treated the same as individuals with GED's because their attrition rates are comperable (which isn't a good return on the investment in time and money to train them). It looks like they look more favorably on homeschooled kids if they do a semester-worth of college. From what I can tell, homeschoolers can get it, but they have to have higher test scores than the kids that graduated from traditional high schools.
http://www.cna.org/documents/D0009351.A2.pdf
It kind of makes sense. If you go to a regular high school, you learn to work in a more formalized environment that you have minimal control over. Your parents enforce discipline, but it's not the same as having to respect and obey someone you don't really know because of their position.
At any rate, good luck with the kid trying to get in. It looks like it should be doable....just a little more challenging. If the army is struggling as much as the MSM claims to meet recruiting numbers, you'd think it should be doable, though. I'm sure they'll be great once they can get in there and show the military their stuff.