erm…
The Army is not having recruiting problems and "strong" is not a adjective modifying the word "Army" in the phrase "Army strong".I believe "Army" is acting as an adverb modifying the adjective "strong" (so I guess the whole phrase is an adverbial phrase). The phrase, which by the way I don't like, is meant to be used as such: "Hey Bill, look at these muscles. I'm not just strong, I'm Army strong."
Someone should tell the experts at Language Log. That post is far from Army strong.
UPDATE: I knew they'd come up with something. Its not an averbial phrase but a adjective phrase.