That's what BCBS of IL calls customer service reps. I would hardly call it customer advocacy. They're nice, but surely they know who pays their salary. And they're certainly not Zappos customer service.
I imagine the theory is that calling someone a customer advocate impresses upon them and maybe the customer that they really are there to work on the customer's behalf, but if employees get mixed messages about their role, terms like customer advocate are worse than useless. If the underlying truth doesn't match the language, then the message is that the company is into smoke and mirrors.
I don't know if Blue Cross Illinois representatives get these mixed messages, but these people represent the company. They can take care of customers and do that well, but changing the language doesn't change the fundamental nature of interests and relationships.