Does an EB-1 "Extraordinary Ability" Green Card applicant have to have an employer?

No. One important advantage of the EB-1 is that (unlike with the non-immigrant O-1 Visa) the applicant doesn't need to have found an employer willing to guarantee a job. In other words, the applicant can "self petition"- that is, apply on his or her own behalf.

This is a critical difference between the O-1 non-immigrant visa and the EB-1 immigrant option.