Barber played college hockey for the Miami RedHawks of Miami University in Oxford, Ohio. In his freshman year, Barber's outstanding play was rewarded with a selection to the 2012–13 Central Collegiate Hockey Association All-Conference First Team. In 40 games, he tallied 15 goals and 24 assists for 39 points and was second in team scoring by then-sophomore Austin Czarnik. On December 22, 2013, during his sophomore season at Miami, Barber was named captain of the United States men's national junior ice hockey team for the 2014 World Junior Ice Hockey Championships. He had been a member of the 2013 United States squad that won gold.