Divers was a creative, goal-scoring but tough inside left, who joined Celtic from Renfrew Juniors in 1932. He helped the club to win two league titles in the 1930s and was also the creative mastermind behind the its victory in the Empire Exhibition Trophy of 1938, when Celtic beat Everton 1-0 in the final. He left Celtic for Morton in 1945, moved to Oldham Athletic for several months in 1947 before returning to finish his career with a second spell with Morton between 1948 and 1949. He earned one cap for Scotland, against Ireland in October 1938.
Divers was the nephew of former Celtic and Ireland player Patsy Gallacher, while his son, also named John, played for Celtic and Partick Thistle.