Enlistment into the US Army is not as simple as this episode makes it out to be, nor is leaving for any military basic training, which is depicted as possibly happening in only one or two days. The fastest it could possibly happen, with a perfect applicant having no history, is five weeks. However, in Ian's case, since he was a long-time participant in Army Junior ROTC and ROTC and has passed several written and physical exams, the pre-enlistment period would be considerably shorter than five weeks but considerably longer than one or two days.
Shameless (2011) is known for breaking the fourth wall before every episode, but Frank also references Showtime while in the hospital.