Last time I checked, there was no concrete regulations governing the admittance of foreign nationals.
Jamesaritchie and Gary said several things that are correct.
Keep in mind that the matter of security will be determined by what occupation the soldier will have. If he's a dirt digger, no problem. He can parents in Russian for all that it matters. If he/she/or it is accessing Confindential level material, still no general problem. After that, at the Secret, Top Secret, etc, that will change.
I've had people Nicaragua, Cuba, Iraq, Iran, Mexico, France, Germany and so on serving in the U.S. military are regular grunts. Generally it's a bit slower for them to get in do to security checks, but it's not an insurmountable problem. To boot, it was all during the height of the Cold War years.
Foreigners may not get stationed at all the cool, secure locations, but they can sure carry a gun and stand a post.
As for loyal oaths........their's carries no more weight than that of an American IMO. We had our own people sell us out plenty of times. The foreign national takes the same oath as do all others.
Hope that helps.