His father, Reuben Oscar Everett, was one of the first five law students at Duke and his mother, Kathrine Everett, was one of the first women to graduate from the University of North Carolina School of Law, where she ranked at the head of her class and was the first woman to argue and win a case before the North Carolina Supreme Court.
Everett was admitted to the North Carolina bar and joined the Duke law school faculty that same year at age twenty two.
[3] During the Korean War Everett joined the United States Air Force, where he was assigned to the Judge Advocate General's Corps.
[3] From 1961 to 1964, Everett served part-time as counsel to the Subcommittee on Constitutional Rights of the United States Senate Committee on the Judiciary, which led to the enactment of the Military Justice Act of 1968.
As both counsel and plaintiff, he twice successfully challenged in the U.S. Supreme Court congressional districts drawn by the North Carolina General Assembly which violated the Fourteenth Amendment of the U.S.