According to The Getty Thesaurus of Geographic Names, there are rivers called Green River in Connecticut, Illinois, Kentucky, Massachusetts, Michigan, Missouri, North Carolina, North Dakota, Oregon, Tennessee, Texas, Vermont, Washington, and Wyoming. There is also a Green River in Canada, but it doesn't say which province.

I remember the local movie theater that I went to as a kid served a soft drink they called Green River. This had to be sometime in the 1950s or maybe even early 1960s, since I would have been only 3 years old in 1950. It was strictly a fountain drink, as far as I knew. I never saw any bottles, and I can't recall seeing any advertising signs, either, or anything that looked "official," and I never saw it anywhere but at that theater. (Of course, I didn't get out much.) I have a hunch it was a homemade concoction. I know the theater also made "cherry cokes" by adding cherry syrup to Coca-Cola, something that probably was not officially sanctioned by the Coca-Cola Company. Maybe they made "Green River" by adding some green syrup to 7-Up. Who knows? I didn't pay close attention in those days.