Greer attended high school in Albertville, Alabama before playing college baseball at the University of Montevallo.

The Rangers made him their tenth-round pick (279th overall) in the 1990 amateur draft, and he spent the rest of the season playing for their rookie-level minor league club in Butte, Montana.

[2] (Rubén Sierra was the first on June 1, 1986, at Kansas City) He also played an integral part in baseball history against the Anaheim Angels on July 28, 1994, when he made a diving catch of Rex Hudler's fly ball to protect a perfect game by Kenny Rogers.

Greer's aggressive style of play took a severe toll on his body, and he spent the remainder of his baseball career undergoing and recovering from surgeries.

He worked out at the Rangers' spring training facility in Surprise, Arizona, with the intention of doing a rehabilitation assignment in the minor leagues shortly thereafter.

However, his workout was cut short; he returned to Texas, where he had surgery to remove scar tissue on his forearm and was declared out for the season.

The Rangers, not surprisingly, decided to buy out the option year on his contract; he actually made almost as much money during the two lost seasons as he did in the rest of his career combined.

Greer received invitations from the Rangers and the Minnesota Twins to go to spring training as a non-roster invitee, but he decided that he would rather retire than move his family to another state.

[8] On July 12, 2023, Greer was named Head Baseball Coach at Fort Worth Christian School located in North Richland Hills, Texas.