Engquist contributed lyrics to such musicals as the Outer Critics Circle Award-winning Kuni-Leml, as well as My Heart Is In the East, Half a World Away and Abie's Island Rose, on which he collaborated with Frank Evans. These musicals were all seen at Jewish Repertory Theater. Among Engquist's other credits are Elizabeth the Queen and Little Ham. His most recent credit was Lorenzo, which was seen at the New York Musical Theater Festival in 2009.
A faculty member at the BMI Lehman Engel Workshop, Engquist wrote a column, "Richard's Almanac," for its newsletter. He also co-wrote Secrets of Good Health with his wife Jane E. Brody.
In addition to his wife, Engquist is survived by their twin sons, Erik and Lorin; a sister, Muriel Cashin; and four grandchildren.