I know the military demands for a candidate which serves as the right fit. But looking at Forrest Gump I think one would have to think maybe thrice!!
He was not mentally challenged or something but he got low IQ:
Image copied from a Quora post covering the same concern and most of the answer there also point out the same thing. And also don't forget he got a football scholarship due to his athletic abilities os, of course, he is quite capable to serve in Army.
Here is all the requirement listed which you need to fulfill to get in the army and it's nowhere mentioned that you need high IQ or even average. So we can assume he did clear all test and got there by fair means.