Eaves attended Shattuck-St Mary's School which is an Episcopal Church-affiliated boarding school in Faribault, Minnesota. He played his collegiate hockey at Boston College. After a very good freshman year the Senators drafted him 29th overall in the first round of the 2003 NHL Entry Draft. He played two more years with the Eagles, where he won several awards, including Hockey East Player of the Year, earned All-America and All-Conference first-team honors, and was a Hobey Baker Finalist. Also during that time he represented his country at the 2004 World Junior Ice Hockey Championships, tallied one goal and five assists in six games. In 2005, he decided to leave Boston College to sign with the Ottawa Senators.