On June 6, 2018, Shugart was selected in the 12th round of the 2018 Major League Baseball draft, with the 370th overall pick, by the Boston Red Sox.
On March 22, 2019, Major League Baseball suspended Shugart for fifty games "after a second positive test violating the drug prevention and treatment program." Shugart subsequently made his first appearance of the 2019 season on May 27 with the Greenville Drive of the Class A South Atlantic League, spending the remainder of the year there.
He opened the 2022 season with the Portland Sea Dogs of the Double-A Eastern League and moved into the bullpen. In early July, he was promoted to the Worcester Red Sox of the Triple-A International League. On August 4, he pitched two innings of a combined no-hitter alongside Michael Wacha and Andrew Politi versus the Durham Bulls. Over 45 games (one start) between the two teams, Shugart went 5-5 with a 5.31 ERA and 63 strikeouts over 62+2⁄3 innings
On August 12, 2024 Boston selected his contract from Worcester. On August 16 he was optioned to AAA Worcester.