Born in Huntington, West Virginia, Greer attended Douglass Junior and Senior High School in Huntington.[1] Douglass was an all-black school. He played as a guard for Douglass' men's basketball team.[2] He enrolled at Marshall University and played college basketball for the Marshall Thundering Herd's basketball team, becoming the first African American to play for a public college in West Virginia.[3]

With the Thundering Herd, Greer scored 1,377 points with a .545 field goal percentage, setting a Marshall record. In 1956, Marshall won the Mid-American Conference championship,[4] and made their first NCAA men's basketball tournament appearance.[5] Greer was named All-Mid-American Conference in 1957 and 1958. He was named an All-American in 1958 as well.[4] Greer finished his Marshall career averaging 19.4 points per game and 10.8 rebounds per game.[6] In 1958, his senior year, Greer averaged 23.6 points per game.[7] Greer also played for the school's baseball team in his sophomore year as a first baseman.[4][5]