Au contraire, mon ami.
Speaking as one whose Daddy couldn't (more accurately, wouldn't have even if he could) pull any strings in that time frame, the options were not exactly as you describe them. For instance, I enlisted in the Air Force (for the minimum four-year hitch) to avoid being drafted into the Army or MC for two. Oddly enough, Uncle Sam then sent me to Canada without even threatening my citizenship (15 months at Goose Air Base, Labrador).
I believe that the two year service periods that you describe were for draftees and not for voluntary enlistments, other than for the AF and Navy, where draftees were rarely, if ever, assigned. Army and Air Natinal Guard enlistments were, indeed, for 6 years.