별일 있어요? means "Is something wrong?", and you can also say "무슨 일 있어요?" It's usually used when you suspect something is wrong with the person you are asking. (or their situation) On the other hand 별일 없죠? sounds more like you are assuming everything is fine, and you are just making sure. Another use of "별" is "별것 다 있어요" , you use when you go somewhere, and are surprised that there is "everything there" including things there that you wouldn't expect. You can use then when your friend is going to a flea market and you tell them, "They have everything there.", possibly including all kinds of stuff that you don't need. A fun, slang use of this "별" that I heard a friend use when we were in a neighborhood with a lot of strange people he said "와~~ 별놈 다 있네" , "wow there are a ton of weirdos here" (emphasizing that there are many different kinds of weird people.) But be careful not to use "놈" unless you are with a close friend. It's a derogatory term.

Lucasme's answer is excellent! for examples, - 'You look serious. 무슨 일 있어요? ( 별일 있어요? (x)) '별' means 'special' 'more than normal'..... It has negative meaning. '별일 없죠?' is sort of "how are you?" "How is it going?", 별일 normally means such as sickness, accident, unpleasant event, etc. Boss can use '별일 없죠?' when he visits his store and say to manager. (Do you have something special to report me?) The next worker can say '별일 없죠?" when he take over from previous worker in rotation work system. (Was everything all right during your work?) I want to explain more and properly but my english doesn't allow me.