It will vary by state, but generally:
When only one of the two parties is unwilling to continue the employer-employee relationship, it is obvious who is the moving party. If employment was still available to the claimant and the claimant refused to continue working, then the claimant is the moving party. If the employer will not allow the claimant to continue work, even though the claimant wants to, then the employer is the moving party.
It's probably not an eviction notice, but a "comply or vacate" notice. Journalists often confuse "eviction notice", which is a court order, with the sort of formal written demand notices that tenants are entitled to receive.