Meyer attended Greensburg High School in Greensburg, Indiana, and played for his high school's baseball team. In 2008, his senior year, Meyer had a 8–0 win–loss record, a 0.95 earned run average (ERA), and 108 strikeouts in 51 innings pitched. Meyer was named a high school All-American, Indiana's Mr. Baseball, and the Indiana Player of the Year by Gatorade and Louisville Slugger in 2008.

The Boston Red Sox selected Meyer in the 20th round of the 2008 MLB Draft. Despite being reportedly offered as much as $2.2 million as a signing bonus, Meyer chose not to sign with the Red Sox so that he could follow through on his commitment to attend the University of Kentucky to play college baseball for the Kentucky Wildcats baseball team. In 2011, his junior year, Meyer had a 7–5 win–loss record with a 2.94 ERA, and led the Southeastern Conference with 110 strikeouts.