Without reference to a particular gun, yes, U.S. weapons were brought back (the real word is "stolen") by GI's. That it was routine and condoned is simply not true. Many people confuse U.S. materiel with captured guns, which were brought back legally (except for automatic weapons). Contrary to the war stories, combat captured long guns are rare, since no soldier was going to carry two rifles with him. Pistols were usually kept if possible. Most rifle "bringbacks" were taken from depots or from piles of captured materiel as the soldiers were getting ready to return to the states.
But U.S. and allied guns were off limits. Of course, in combat, some were lost, weapons were swapped, picked up, and so on. But even if a soldier did, for example, lose his carbine and acquire an M1, it was still not his to keep.
In many cases, vets acquired military guns after the war, when surplus arms were cheap and plentiful; even if the vet was honest (some may actually have been), "a gun like dad carried in the war" soon became "the gun dad carried in the war."
Two tales I have told before are illustrative. The widow of a man who was a captain when he landed on Omaha Beach showed me the carbine he carried that day. If it was, it was the only Model 1873 carbine on the beach.
The second is about a young man who told me his late father had an M1 rifle that was given him by Gen. Patton for heroism at the Battle of the Bulge. But I had known his father before the son was born; he had never been out of the states, and I was with him when went to the Railway Express office to pick up the M1 he got from DCM.
As I said, some "bringbacks" may have been that. But I know of too many that were not, that I consider it foolish to pay any more for them based on unsubstantiated stories.
One point to consider. If you were a soldier, coming home after a rough couple of years in a combat zone, whether in actual fighting or not, would you risk ten years in Leavenworth just to bring back a weapon you were probably heartily sick of anyway? Some did, but not as many as we tend to think.