Wow, loaded question. You are probably quite used to shoes, and have conditioned your ankles accordingly. Might not be the case for all of the boys.
Seems that mountaineers are the only ones who absolutely need the classic heavy leather (or plastic) boots. How heavy are your packs? Might want leather boots for over 40 to 50 pounds, synthetic boots for over 30 to 40 pounds, and shoes for anything less.
For the boys, perhaps leather offers a little better safety margin given the possibility of side impacts, twisted ankles, etc., but if you teach them about the advantages of light footwear and caution them what to watch for, they can probably be successful with shoes too (within the aforementioned weight ranges).
Also, there is quite a bit of difference between shoes. Shoes designed as trail runners or hikers tend to be much more stable and supportive than road runners. Basketball, tennis, or casual shoes really have no place on long hikes. I haven't looked at any Nike AGC's lately, but some in the past were relatively heavy and not particularly supportive. Currently popular shoes are Salomon, Montrail, Vasque, Adidas, Inov8, Keen, North Face, etc.
For you, I think I'd choose the shoes, or at the very least, some much lighter boots. But... I've never been to Philmont - perhaps the trails are full of razor-like scree?!?
Oh yeah, one final note. In some conditions I use short stretchy gaiters with my trail runners. Good for keeping dirt, seeds, and other debris out of my shoes. Mine are REI Mistral Gaitors.