The Mets also signed right-handed pitchers Chase Ingram (sixth round), Corey Taylor (seventh round) and Witt Haggard (10th round), catcher Patrick Mazeika (eighth round), and left-handers Sixto Torres (17th round), Taylor Henry (21st round) and Seth Davis (29th round).
The Mets' top overall pick, second-round outfielder Desmond Lindsay, is not yet signed. But scouting director Tommy Tanous said last week that the club fully expects to ink him.
One of two high school left-handers the Mets selected with their first four picks, Szapucki ranked 97th on MLB.com's Top 200 Draft Prospects board. Tanous called him "a lefty with incredible, really good arm strength and stuff."
Anthony DiComo is a reporter for MLB.com. Follow him on Twitter @AnthonyDiComo and Facebook, and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.