If an employee is sexually harassed and they go to H.R, are they protected by silence similar to client privilege?

No.  Most employers will try to keep as much of the information as possible confidential.  However, when an employee reports sexual harassment the employer has an obligation to investigate the report and determine whether harassment has occurred.  In order to do that, the employer will have to interview any witnesses, and in doing so will probably reveal information which will reveal who is making the complaint.  If a particular person is being accused of the harassment, that person will have to be interviewed.  During that interview the identity of the person making the complaint will probably become known to the person being accused.  Otherwise, the person being accused cannot be given a fair chance to respond to the complaint.