Enright was the son of James E. Enright (March 1871 – ?) and May C. Billick (January 1882 – ?). He was born in Rockford, Illinois. He graduated from Central High School in Rockford, where he played on the 1918 state champion basketball team, and was team captain and played running back on the football team. He then graduated from Notre Dame University where he played running back for Knute Rockne on the football team, and also played on the Irish basketball team. He married Alice ? (1903 – ?), they had daughters Eugenia M. (born 1927), Alice E. and Joyce. Enright played fullback for the Green Bay Packers in 1926 and 1927.