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 the University of Notre Dame 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.
In the early 1930s Enright was an assistant football coach at the University of North Carolina at Chapel Hill. From 1931 to 1938, he was the head basketball coach at the University of Georgia and assistant football coach. In 1938, he was hired as head football coach and athletic director at the University of South Carolina. After the 1942 season, he joined the United States Navy serving as a lieutenant and working mostly in their athletic program in the United States. He returned to the Gamecocks in 1946 as head football coach succeeding John D. McMillan, and remained until 1955 when he resigned for health reasons. He hired Warren Giese as his successor, and continued as athletic director until 1960. The Rex Enright Athletic Center on the South Carolina campus was named for him and the Rex Enright Award (also known as the Captain's Cup) given to the football captains of the previous season. He was considered one of the "ring leaders" in the formation of the Atlantic Coast Conference in 1953. He is a member of the South Carolina Athletic Hall of Fame. He died in 1960 of peptic ulcers and a rheumatic heart.
In 2009, the University of South Carolina recognized Enright as the winningest football coach in school history. His record 64 wins was highlighted during the halftime show of South Carolina's football game versus Florida Atlantic on September 19, where Enright's daughter, Jean Smith, and great-grandson, Brian Garrett, accepted the presentation in his memory. However, that record lasted only two more years when Steve Spurrier passed him in 2011.