It's a quirky allusion to the idiomatic square peg in a round hole, used to refer to a situation where someone is not1 well suited to their circumstances.
Which is almost always that way round - people don't normally talk about being a round peg in a square hole.
But Gump is saying that he is a metaphorical "round peg" - with the implication that for most people, being in the Army is like being a square peg in a round hole, so they don't fit in. But Gump does fit in (being in the army is weird, but Gump is weird too, in a compatible way).
In case it's not obvious, the specific type of "weirdness" alluded to above is essentially that both Gump and the army in general are stupid, unimaginative, literal-minded.
Here's another example that more explicitly clarifies the intended sense of being "a good fit":
I had a very happy early experience in policing in that my first two years everything fitted into place. The concept of being a round peg in a round hole. I was that round peg. This was my career.
1 The established idiomatic usage refers to someone who doesn't fit in. Where Gump's "quirkily reversed" version means he does fit in (specifically with army life, not necessarily life in general).