Yes, they can. With very rare exceptions that do not apply here, no employer in any state is required to confirm (or, for that matter, deny) previous employment. It's a mean and petty way to respond, but it's legal.

She can file a wage claim with the state DLSE.

Good luck on her unemployment appeal, but don't expect too much. The definition of "good cause to quit" and still receive unemployment is VERY high.