The Bruins (37-15-8, 82 points) are one point behind the Toronto Maple Leafs for second in the Atlantic Division and five behind the first-place Tampa Bay Lightning.
Nash is in the final season of an eight-year contract and can become an unrestricted free agent July 1. He had 28 points (18 goals, 10 assists) in 60 games with the Rangers this season and has 799 points (434 goals, 365 assists) in 1,049 games in 15 NHL seasons.
"It was a strange last few weeks in New York, but I'm moved on," Nash said. "I'm excited for this opportunity and excited to be a Bruin."
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Rangers coach Alain Vigneault, who called Nash one of the best 200-foot forwards he has coached, inserted Spooner in the lineup Sunday. The 26-year-old had two assists in a 3-2 overtime loss to the Detroit Red Wings.
The Rangers have lost seven consecutive games and are last in the Metropolitan Division, seven points behind the Columbus Blue Jackets for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
Spooner has 27 points (nine goals, 18 assists) in 40 games this season.
"What I see is a real good skater, good hands, very good on the power play," Vigneault said. "He can play all three positions. … We'll give him a good look here and see what he can do, but everything I've heard has been real positive."
Beleskey, who has been playing for Providence in the American Hockey League, will remain in the AHL and play for the Hartford Wolf Pack, the Rangers' affiliate. Lindgren is in his sophomore season at the University of Minnesota.