It would just freeze, but since your out in the middle of no where just pee on the ground, what really sucks is if your the guy that's gotta pack the "stuff" in the bucket with you because the ground is so frozen you can't bury it...
No, they keep these buckets inside their houses and cabins. Usually in the corner. Most of the time the crap buckets are the ones with the turtles in them. Some of these Alaska villages are still living by third world standards. No jobs, no money.