Shall we summarize this?
They are breaking the law. They probably do not agree with the law but they are breaking it. NY, since there is a registry, can enforce this law and is choosing to. There are very few options once you receive that letter because the people sending it are the only ones with the legal use of force backing them up. They can and will screw you over if you do not comply.
I find it interesting that NY jumped straight to enforcing the "Collect the bad guns" instead of the way CA did it with "Get gun from illegal owners." Perhaps because there was so long with restricted access the only people on the registry any more are legit owners of now illegal guns. This enforcement action will do much more to convince people against registries than what CA is doing; collecting .22 bolt rifles becasue they hold 6 shells is unlikely to garner as much sympathy as collecting .45 handguns from people under restraining orders.