Do we have any rights in customer service industry?

It stinks, but sometimes you have to be better than the person on the other end of the line. My sister was fired from her job at a call center for disconnecting a person who was cursing at her(she does not use profanity at ALL). I think letting them vent and then explaining your answer or giving them an answer is the best way. If they don't respond, bring in a manager. You're not there to be someone's punching bag, but it happens. If you flip out or disconnect them, they'll most likely can you. Good luck.