Personally, I would not go after black bear with a .357. Black bears are very aggressive and given the opportunity, will not retreat and they will kill you.
I have to disagree a bit with this, based only on my personal experience with black bears. Have met a few over the years, and I found them as individual as the people I have met. One might be agressive (or just hungry) while another might just want to be two counties away from you as fast as he can get. Mama bear with cubs, on the other hand is a completely differrent beast.
I grew up hunting in the Adirondacks, and the bears in the northeast are not gigantic armor plated monsters. In fact, they seem much bigger than they really are. Most of the time, stories about bears taking a lot to kill are not because of any lack from the gun, but from the shooter, not putting the bullet in the right place. And thats beacuse bears are smaller inside that heavy bear coat than they look. So the vital area isn't right where it would seem to be. Close, but something that you need to study a bit to ensure accurate and effective bullet placement.
When I left the northeast in the mid 1970s, the record black bear (and the biggest taken in a century) was a monster 600lb. That's a freakishly huge black. Usually they only run half that size, or even less.
Old timers when I was a kid, thought a .30-30 was fine for deer, and a .32 Spl for bear! .35 Rem was highly thought of too.
Hunting bear is not quite the same thing as defending yourself from bear. The .357 from a full length barrel, with good solid or jsp bullets will do fine for hunting. 158gr would be my choice, as I don't care much for the 180s. Do not use the 125gr bullets, they are optimised for use against people, not bears.
And don't shoot the bear in the head. I have personal experience with a black that took several .30-30 rnds to the head (at short range), and was not put down by them (although he was rather confused). It had nothing to do with the power of the gun, but the rounded shape of the bear skull. None of the bullets penetrated, they glanced off. A couple rounds through the chest dropped the bear easily. The head shots should have done the job, and on another day likely would have, but that day, things just didn't work like expected. It happens.
Its not a popular opinion now days, but the .357 will do fine for deer as well. Somehow, everybody seems to think you need the biggest guns and the heaviest bullets, but that has not been my experience. Deer can be amazingly tough appearing, taking several chest shots to drop, sometimes. Other times, they drop like lightning from a single hit in the same spot. All depends on the deer, as far as I can see.
Hunting in the New Hampshire woods, long shots are not common. If you are good enough, your .357 will do just fine, and if you aren't, no monster magnum will make up for that.