After practicing with a private law firm in Madison, Wisconsin, Eich served as an Assistant and Deputy Attorney General of Wisconsin with Attorneys General Bronson La Follette and Robert W. Warren.

In 1971, Governor Patrick Lucey appointed Eich as Chairman of the Public Service Commission of Wisconsin.

Lucey later appointed Eich to be a Judge of the Wisconsin Circuit Court in 1975.

Eich was appointed to the Court of Appeals by Governor Tony Earl in 1985.

Since retiring from full-time judicial service, he has remained a reserve judge for the Circuit Court and served as a legal advisor to the Wisconsin State Journal.