I had an instructor in AIT who assured me that the Army of One slogan was actually an acronym, with One referring to "Officers, NCOs and Enlisted." But this was one of those instructors whose words you had to take with a grain of salt, so I'm not quite sure of the validity of this. Still, it makes more sense to think of it that way than to think that the ad campaign was trying to appeal to the selfishness of young people, which, in my opinion, does a great disservice to what the Army is all about.
(If it was an acronym, you'd think the good people at the marketing company would have spelled it as O.N.E. instead.)