Hi there. Former swab, now rising 3/c. I don't particularly remember what my forms said (honestly this whole process will be a blur by this time next year to your cadet too) but honestly you can get away with any shoe. I don't recall there being a color specification, which is weird because the color of your running shoes is the one thing you can get away with having any color of (although one guy in my company was nicknamed Flash for his bright yellow shoes.) I wouldn't stress about it, as your swab should definitely invest in a nice, comfortable pair of running shoes (also some nice shock-absorbent inserts.) You will be issued shoes that they'll wear in their normal uniform, but for PT you will want nice shoes because you're waking up 6 out of 7 days a week to do a morning work out and you're running everywhere. In addition, these shoes will be worn when they do the O-Course.
I would also invest in a good pair of high-rise tube socks like said above because they will carry their running light and pen in them, and you do not want short socks or something your running light would fall out of -- trust me, the cadre will have a field day with your swab if they do! Good luck.