Both "stand" and "bear" work. However, it sounds more correct to say "I can't stand/bear working the late shift" or, "I hate working the late shift." Also, if by "late shift" you mean late afternoon and night, we usually call that the "swing shift".

I can't stand to work the night shift. It means you can't tolerate or dislike the situation. I can't bear to work the night shift. This would be the same as the last sentence, but in diffrent meaning. Like you can't stand the fact that your working.