Obviously, none of us know for sure, but I really suspect that Boyle is a huge, huge part of the team in terms of setting the tone on the ice and in the room. Befriending new players, bonding with teammates etc.
I don't think they should match any UFA offers, but I would definitely sit him down and explain the situation to him. NYR want to keep him, but they're building a winner in the short term, and they can't afford to match desperate bottom teams' UFA offers. If he wants to take a number that isn't built on what those bottom teams offer, they can/should/will work something out, and he can stay here in a role that he is great at and try to win with the guys he's been playing with for many years in some cases. If he wants to take the highest number, that's fine, but NYR can't and won't try to match it. It's been a good time, but it is what it is.
Hard to say how it'll shake out. On one hand, Boyle seems to really like it here, and I wouldn't be shocked to see him leave some cash on the table to stay in New York. On the other hand, he's not a 5 mill/year guy, and leaving money on the table may really affect his quality of life down the line, way more than it would with a Callahan level player.
Either way, I've loved Boyle here. The only times he was frustrating were when he was asked to do too much.