My employer is asking for a copy of my criminal history record. Can I give it to them?

No. 

California Penal Code section 11142, prohibits you from giving your copy of your criminal record to an unauthorized third party. 

In addition, California Penal Code section 11125, prohibits an individual or agency from requiring you to provide him/her or the agency with a copy of your criminal record or proof that a record does or does not exist. Violation of either of these sections is a misdemeanor offense.