The 32-year-old MacDougal had a 12.46 ERA with seven walks over 4 1/3 innings with Chicago.
Egbert, 25, will work out of the bullpen. He was 1-0 with a 6.43 ERA in five appearances with Charlotte. The right-hander is 40-36 with a 3.35 ERA and 595 strikeouts in 134 games (including 113 starts) over five minor league seasons since being selected by the White Sox in the 13th round in 2004 draft.
Egbert has not yet pitched in the major leagues.
The moves, which were made before the White Sox faced the Baltimore Orioles, reduced Chicago's 40-man roster to 38.