I wouldn't dream of using any handgun less than .357 Mag for whitetail.
And I stopped using 12 GA for squirrels around college age because it was a bit too much and really too easy. Same goes for semi-auto shotguns in general with a possible exception for birds, but even then I wouldn't use it. More of a challenge to harvest squirrel with .22 LR or a single barrel 20 GA.
I wouldn't bother with something as beastly as .375 H&H for anything in North America short of Kodiak.
Mauser Werke, Schmidt-Rubin, Colt, Walther, HK, Weatherby, Browning, Ruger, Beretta, Savage, Bushmaster, Marlin, Mossberg, Remington, PTR, Saiga, S&W, Sig Sauer....a few friends of mine