After graduating from Fairfield College Preparatory School, Drury was drafted by the Quebec Nordiques 72nd overall in the third round of the 1994 NHL Entry Draft. Upon being drafted, he began a four-year career at Boston University (BU). In his freshman year in 1995, he won a national championship with BU. In both 1997 and 1998 he was named Hockey East's player of the year and was also named best defensive forward in 1998. In his senior year, he won the Hobey Baker Award as the National Collegiate Athletics Association (NCAA)'s top ice hockey player after finishing as the runner-up the previous season. Drury was the first BU player to reach 100 career goals and assists, finishing with 113 and 100, respectively. BU won the Beanpot tournament all four years he played for them. On January 15, 2009, well into his NHL career, Drury was named Hockey East's Best Defensive Forward, as part of the league's 25th Anniversary celebration. Drury was chosen in a vote of Hockey East fans and members of the league's 25th Anniversary Committee.