the OP did not ask what "would" you carry he asked what "do" you carry. Not sure many can answer that in the shotgun section.
While I have done a considerable amount of backpacking, carrying a shotgun has never been on my list. On several occasions when deer hunting was slow and squirrel was in season I would pack around an old Snake Charmer with 4 rounds in the stock. I would not consider doing this on a backpack trip though. On some of my backpack trips where we knew we were going to be in cottontail rich areas we would plan a meal with fresh rabbit and I would take my old AR7 but that is a 22.
While I have done a considerable amount of backpacking, carrying a shotgun has never been on my list. On several occasions when deer hunting was slow and squirrel was in season I would pack around an old Snake Charmer with 4 rounds in the stock. I would not consider doing this on a backpack trip though. On some of my backpack trips where we knew we were going to be in cottontail rich areas we would plan a meal with fresh rabbit and I would take my old AR7 but that is a 22.