When a massive data breach struck retailer Target during the midst of the 2013 holiday season, consumers responded by shopping at other stores and its brand was briefly tarnished. Now, Anthem, the nation's largest Blue Cross operator, finds itself in a similar position. But switching health insurers is a whole lot more complicated than heading over to another store.

Anthem said it discovered the data breach a week ago, and its IT team was able to contain it and quickly notify the Federal Bureau of Investigation. But the damage could be devastating for as many as 80 million past and present customers and employees of the nation's second-largest insurer.