Abner attended high school at Mechanicsburg Area High School in Mechanicsburg, Pennsylvania, where he was identified by New York Mets scouts as an all-around "can't miss" outfield prospect after his sophomore year. He was also a star at football, and his number "16" was retired by the school for use in both sports. The Mets made him the first pick in the 1984 amateur draft, having had recent draft successes with Darryl Strawberry and Dwight Gooden, two players who became stars almost immediately despite their young age.
Abner never played a big league game for the Mets. After moving slowly through the minor league system, he was traded after the 1986 season to the San Diego Padres as part of an eight-player swap that sent Kevin Mitchell west in exchange for Kevin McReynolds. He would make his debut on September 8, 1987 in a loss to the Braves. Abner entered in the top of the eighth inning as a pinch hitter for Lance McCullers and flied out to center field in his first major league at bat.
The light-hitting Abner played sparingly over the next five seasons for San Diego, being used mostly as a reserve outfielder. He would be traded to the California Angels in 1991 and signed by the Chicago White Sox in 1992. His season with the White Sox was probably his best, as he attained a .279 batting average over 97 games.
However, Abner would injure his knee the following season playing basketball before a Triple-A game for the Omaha Royals, and never played in the majors again.