No, you do not have to let them. If this is a water softener for the entire home and it is built into the water lines then it is definitely part of the home unless stated in the contract as a "none included" item.
I think this is a situation that should be dictated by what is right, not what rights do they or you have. If you did not think this item was part of the sale then maybe give it back. This is a short sale and these people have lost their home. Maybe they don't need anymore hardship. Just a thought. These are tough times for many and if we all showed a little compasion to our fellow man the tough times would be less tough.
Answer? No. You really don't have to let them have the water filter. But why would you not want them to have it? Were they nasty to deal with? Or, is it just a matter of principal? Frankly, unless they gave you a hard time and you just want to get even, I'd let them have it. It's so incidental in the overall scheme of things and you'd probably be better off with a new one anyway.
Philosophically it's kind of a Karma thing. If it makes you feel good keeping something that means something to someone and doesn't really mean anything to you then I guess it's yours. Good luck on this one and enjoy your home and all its contents.