Swayman was drafted by the Bruins in the fourth round (111th overall) in the 2017 NHL Entry Draft, and at the time, he had already committed to playing college hockey at the University of Maine.
In three seasons with the Black Bears men's hockey team, Swayman appeared in 100 games, with a total record of 47-40-13. He finished his college career as Maine's all-time leader with 3,130 saves and 5,906:45 minutes played, while his 2.51 goals-against-average was fifth in franchise history, and his .927 save percentage was behind only longtime Detroit Red Wings goalie Jimmy Howard.