It's definitely an American thing, but any U.S. students who had classes discussing Vietnam/the '70's controversies almost surely heard it at some point. That's where I first heard it, but it was based on Thomas Paine's famous pamphlets from our Revolutionary War, where he discussed the idea of the "summer soldiers," the soldiers who are just bandwagon jumpers who would desert as soon as the tide turned or things just got too rough. The term "winter soldier" was coined as the opposite of that - the guys who would hold firm to their convictions even if the bandwagon was going the other way. Our current Secretary of State, John Kerry, helped make the term famous back in the '70's when he gave an historic testimony as a Vietnam vet, revealing war crimes committed by our own troops over there. The documentary you referred to was made shortly after that time about the investigations surrounding this ordeal, but that's not where the term originated from.
Given the definition of the term, the significance it holds in our country's history, and how perfectly it fits Caps main plight in this film (having to stand against our own government and expose its crimes), it would take 
a lot to convince me it's a mere coincidence, writer confirmation or not.
EDIT: Found an article in which Ed Brubaker confirms that the name is a reference to Thomas Paine and the Winter Soldier Investigations of Vietnam: 
http://www.huffingtonpost.com/william-bradley/winter-solider_b_5060177.html