Yes, I know that. But, just being a lawyer doesn't automatically mean that you have an attorney-client privilege with every person you know.
In order for Mills to claim attorney-client privilege with Clinton, she has to represent Clinton as a client. But, Mills is a State Dept. employee. Representing Clinton and being an employee of the US government at the same time, especially pertaining to US government business, appears to be a serious conflict of interest.
The only think I can think of is that the Clinton-Mills relationship in the past was as an attorney-client even when they were both employed by the state department.