Well, OK, if it absolutely comes down to just one gun, then:
  1.  If you plan to walk long distances with stuff on your back, then a .22LR, preferably in semi-auto. (Ruger 10/22 or Remington Nylon 66).
  2.  If you don't plan to walk long distance, then a Mini-14 Stainless in .223, with a .22LR conversion kit.
  I have owned and hunted with shotguns for over 40 years, and quite frankly, I am not that impressed by them.   I would never choose a shotgun as the One-Gun solution.  The ammo is way too heavy, bulky, loud, and short-ranged for what you actually get in terms of effectiveness.
Regards,
Mannyrock