Burger, a third baseman from Missouri State University, was taken 11th overall in this month's Draft. The 6-foot-1 junior led the Bears with a .328 average, while hitting 22 homers and driving in 65 runs in 63 games.
Sheets is the son of former Major League outfielder Larry Sheets, who spent most of his eight-year career with the Baltimore Orioles. Gavin Sheets starred at Wake Forest as a first baseman and hit .317 with team highs in home runs (21) and RBIs (84).
Of the nine Chicago picks that remain unsigned, four are still playing in the College World Series: Lincoln Henzman, Kade McClure and Logan Taylor of Louisville, and TCU's Evan Skoug. Henzman, McClure and Taylor were taken in the fourth, sixth and 16th rounds, respectively, while Skoug was Chicago's seventh-round pick.
Tyler Mason is a contributor based in Minneapolis who covered the White Sox on Thursday.