During his childhood, the family moved to Columbia, Mississippi, where his father started a dental practice and his mother ran a Christian bookstore.
At the age of twelve, Bush made a confession of faith and was baptized into the First Baptist Church of Columbia, Mississippi.
He went on to attend Mississippi College where he met Cynthia Ellen McGraw, whom he married on June 2, 1968.
Bush first joined the faculty of Southwestern as a teaching assistant while pursuing his doctoral studies, and continued to serve as professor of philosophy of religion from 1973 until 1989.
From 1989 until his retirement in 2006, he served as academic vice president and dean of the faculty at Southeastern Baptist Theological Seminary.