I am assumed the writers wanted a patriotic reference/connection to the bald eagle as the US's national bird. It could also be a play off the expression "soar like an eagle" or "fly like an eagle." It is my understanding that both expressions mean that the "eagle" was meant to soar/fly in the heavens instead of remaining on the ground with the chickens. In a lot of ways the movie is about Doug realizing he is meant to be a fighter pilot (aka the eagle) and being able to fulfill it by the end of the movie. Also, "Iron Eagle" seems to roll off the tongue slightly easier than "Iron Falcon." Also, I think the eagle is viewed as a more superior predatory bird than the falcon (most likely because of the size difference between the species).

FWIW Iron Falcon seems to be a fantasy roleplaying game, so there may have been copyright issues with the name.