The title is a reference to G.I. Joe. "Grunt" is United States military slang for an infantry soldier. 'Annoyed grunt' is the script notation for Homer's favorite exclamation, D'oh. Essentially, this title is G.I. D'oh!
When the colonel has the recruits do pushups while Homer eats donuts, it is referencing a scene in the movie Full Metal Jacket. The movie is again referenced when Homer wonders if the drill sergeant will ask him "what his major malfunction is".
The scene with the predator drone (the mini-helicopter) chasing Homer is a parody of a typical Looney Tunes chase scene. Homer fools it by pretending to dress up as a female predator drone, using soap suds to disguise himself as an old man taking a bath, upon seeing him like this, the predator drone holds up a sign saying "Sorry, sir!", and finally, Homer lures him into a closet filled with a ridiculous amount of TNT. The sequence ends with a "That's All Folks!" logo with The Three Stooges theme song.
When Homer explains why they shouldn't surrender to the Colonel's troops, he mentions that the Army will strip people naked, put a bag over their face, and have hicks laugh at them (as well as Cletus recognizing the description as matching a photo of what Brandine did while in Iraq), referring to the torture controversy at Abu Ghraib Prison.
When Lisa attempts to lecture the Army Colonel about how an occupying force is outmatched by determined people in a foreign nation, she cites the Vietnam War and its result as one of the sources.
When the Army rounds up all men who are bald, fat, or have been amused by the antics of Homer, Seymour Skinner is among the captured men (despite not being bald or fat). Principal Skinner could have been amused by Homer's tomfoolery.
When the army comes into Springfield, Lenny is crushed inside his car, and seriously injured, but he is seen drinking at Moe's in the next scene. This was most likely a time skip between the two scenes.