Why couldn't Greenway be mentioned with the current crop of exquisitely talented American NHL players such as forwards Auston Matthews (Toronto Maple Leafs), Jack Eichel (Buffalo Sabres) and Clayton Keller (Arizona Coyotes), and defensemen Zach Werenski (Columbus Blue Jackets) and Charlie McAvoy (Boston Bruins)?
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Quinn has told him he is in that group. But, Quinn said, "He doesn't really believe it. I think he's starting to."
Part of that is the phone call that Greenway received a couple of days before Christmas, inviting him to play for the United States men's team at the 2018 PyeongChang Olympics.
One of four college players on the team, Greenway will also be the first black player on the U.S. men's Olympic team. He will break that color barrier 60 years after the first black player, Willie O'Ree, entered the NHL.
It's not something he ever expected; he didn't even know the barrier had yet to be broken.
"I'm honored," Greenway said. "I think it's unbelievable. At first I thought I was just another kid going to the Olympics, but now I think it gives me the opportunity to maybe be a role model to other African-American kids who are trying to figure out what they want to do.
"I think a lot of kids have other role models in other sports, like basketball, baseball, football. Because there's not as many African-Americans playing hockey, hopefully I can be one of those role models and maybe influence them to come out and play hockey and try something different.
"Hopefully I'm the first of many."
The best times were the mornings, 5:30 a.m., 6 a.m., when the Greenway brothers would wake up and head for the ponds. Jordan and J.D., younger by one year, had started on skates at 3 and 2 years old, and knew their aunts and uncles always kept their ice surfaces nice and smooth, places for them to go by themselves or gather their friends in a town that was basically Canadian, 15 minutes from the border, and almost completely white.
"We'd play all the time," Jordan Greenway recalled of growing up in Canton, New York. "It was crazy. My family did everything they could to get us to whatever rink we wanted to, so [when] we grew up literally 24/7 we were playing hockey."
It wasn't always easy. Their father was not in the picture, but the family was always there, those aunts and uncles, ready to provide ice or support or rides to hockey games. Theirs was a hockey family, the sport passed down. It seemed inevitable that it would reach Jordan and J.D.
"I remember my great-grandfather, he was always into the [Ottawa] Senators and he always brought us to games when he could," said J.D., a sophomore defenseman at the University of Wisconsin. "I think just something about the sport captured my family, and we just continue to pass it down to the younger generation.
"Thankfully it reached my brother and me. I just think it's our job to continue passing it down."
But for now, it's their job to play.
It's exactly what Jordan has done this season, making strides in his game that support his decision to return to BU rather than head to the Wild. That will likely come next season, with Greenway having all but decided to sign a contract and play in the NHL.
Quinn has moved him to center with the idea that it will push Greenway to be better and more involved in the play, something that has not always come intuitively to him. Playing center forces him to be more alert, with more responsibility.
Greenway's big body (6-foot-5, 230 pounds) has allowed him to hang onto the puck for too long, has prevented him from having the quicker release that he can and should have. The plan is for him to work on playing against his body, to protect less and shoot more.
"There had to be more consistency in his play," Quinn said. "As good as he's been for us, there's been a lot of peaks and valleys and the valleys have been way too low, and he needs to iron that out.
"He's a great penalty-killer. He's got great vision and hands. He can skate. He should be a much better goal-scorer. I'm shocked he hasn't scored more here."
In three seasons at Boston University, Greenway has 82 points (24 goals, 58 assists) in 104 games. The question is what he can do in the future, at the Olympics and with the Wild.
The former is more pressing. The United States opens play Wednesday against Slovenia. From there, it's not only what Greenway can do to impact the team. It's what the team as a whole can do.
"I am confident," Greenway said. "I've got to be, right? I'm not going to be, we're going over there for bronze. We're going over there for a gold medal."
Confidence is not usually Greenway's strong suit. It's not who he is, but it's something that Quinn and company have tried to draw out of him during his time at BU. They have seen improvement. They have seen inklings of that person blossoming.
It shows in his dealings with the media, which have become more numerous in the wake of his selection to the Olympic team, and in his play. They have seen it in his feelings about next season, when he expects to play for the Wild -- though no one is ruling out a return to BU, given how comfortable he is in that environment.
"I definitely think I could step in and play in the NHL next year," said Greenway, who turns 21 on Friday. "I'm confident that I could. There's definitely things that I'm going to learn. I'm going to make a lot of mistakes, but I think I'm confident enough to be able to go there and play and I think that soon enough I'll have a pretty big impact there."
That's a change in itself.
"He's become more assertive," Quinn said. "He's grown up away from the rink.
"He needed college, he needed the structure. I'm not just saying BU, he needed college. He needed to learn. There's an awful lot that goes into being a pro, and so many guys don't make it -- not because they're not talented enough, because they don't know how to make it."
Greenway is learning. He is getting better. He is as close as he's ever been, perhaps weeks from signing with the Wild, perhaps months away from making his dreams of playing in the NHL come true.
Greenway is more mature now, more understanding of what he has and what he can be. He is working to align his nature with the nature of his position and the nature of hockey, and through that, find his place in the game.
"Some people are born arrogant," Quinn said. "Some people are born confident. Some people are born insecure. I just think that's him. It's just ho-hum Jordan Greenway. I tell him all the time, 'Your problem is you've got to fight your personality because you're a very laid-back, easygoing, fun-loving guy."
It might be working. Greenway might, finally, be seeing himself for the player he can become, for the player he is.
When Greenway was playing in the IIHF World Championship last year, he placed a call to Quinn. It was a week into the tournament and Greenway wasn't playing as much as he would have liked. But that hardly mattered, not for what Greenway was quickly realizing.
"I'll never forget," Quinn said. "He's like, 'Coach, you know how you've been telling me this whole time that I should be the next great American? I can play in this league. I think you're right.'
"I think he's starting to realize how good he can be. He can do everything. There's nothing he can't do. It's all up to him."