After playing prep school hockey in New Hampshire, Greer was offered a spot to skate for Boston University starting in 2014 and played for the Terriers for parts of the next two years.
In his first season in Boston, Greer played in 37 games as the youngest forward in the NCAA and went on to register three goals and four assists. BU had a strong year and made it to the Frozen Four. On the biggest stage of college hockey, Greer scored a key goal to help the Terriers to a 5-3 win over North Dakota in the national semifinals, which gave the club a chance to play for a championship two days later. In the national title game, the Terriers put up a fight but fell 4-3 to Providence.
His strong freshman campaign at BU helped propel him in the draft rankings, and the Avs used the No. 39 overall selection on him that summer at the 2015 NHL Draft.
During his second season at BU in 2015-16, Greer played in 18 games and registered five points (one goal and four assists) before deciding it would be best for his development to depart BU and pursue an opportunity to play in the Quebec Major Junior Hockey League with the Rouyn-Noranda Huskies.
"I think BU is a great program, great facilities, great coaches," Greer said to ColoradoAvalanche.com. "It just wasn't the right fit for me from a development standpoint. It was a tough decision to leave a diploma hanging, but I had to make a decision for my career and I wanted to be a hockey player."
The risk of giving up his NCAA eligibility and putting graduation on hold was large, but it was a risk he was willing to take.
"Obviously, there is leaving a diploma from one of the best universities in the world, and another thing was if it doesn't work out in hockey, I don't have that backup for school," said the Joliette, Quebec, native. "School is very important for myself and my parents, so whether I play hockey and it doesn't work out or something, I can always go back to school. For hockey, it was a big risk. It was a make or break situation. I just gave it my all."