These sentences are different in what they are suggesting. 'This can't be happening' means that you don't believe what is happening, whereas 'This shouldn't be happening' means that you think that what is happening is wrong, or not going according to plan.

Hi, Wind! I'll agree with you on this one. They have very similar meanings to me, but "shouldn't" is a bit weaker in my eyes than "can't". "This can't be happening" is used when you, beyond a shadow of a doubt, think that something have 0% chance of happening, but it still is. "This shouldn't be happening" means that you knew there was a chance of it not happening, but you also thought it could possibly happen. This is purely my opinion on how I view these two different sentences, and how I would use them. Hope this helped! -Kyle

The second one, with "can't", indicates dismay. The first one, with "shouldn't" means serious concern. The first expresses a stronger emotion.