Per this story
As they do not plan to recover it, it is abandoned - I suppose you might have a shot at keeping it. It seems to me that it falls under the same sort of heading as a sunken ship. Or at the least they would be required to pay you for recovering it.
Give it a shot and let us know how it works out.
http://www.weeklyfisherman.com/abandonedlost.htm
“This brings us to the concept of an “abandoned” vessel… Such a vessel is legally created when it is determined that there was an intent on the part of the original owner to permanently relinquish title to and possession of the vessel. Such intent can be determined by, for example, making a claim for insurance. But, one thing is usually for certain. The mere passage of time has not be en historically approved by the courts as a manifestation of intent to abandon. It remains an important factor, but, standing alone, is not sufficient to prove the vessel abandoned. In cases where ships are deemed to be abandoned, it is a much more simple thing to obtain absolute title. This is determined by the law of finds. If a ship is actually deemed abandoned, title to the ship and its goods passes to the finder. Please note here, however, that the Abandoned Shipwreck Act has changed this concept considerably.”
I recall once reading a site about international law, and it said the same thing. If a ship (or, in this case airplane) is knowingly abandoned by its owner, than the owner loses title. Anyone finder who wants it and takes it can claim title. Makes perfect sense legally.
And, in this case since NASA has announced they have no intent to recover this aircraft, why would they care if someone else did? At this point, it has served NASA’s purpose, and to them is just garbage.
Considering they didn’t make much of an effort to slow it down before it hit the water, how much of it would be left? You might be searching a rather wide area for pieces of scramjet.
However, doesn’t title then pass to the insurance company?
WAG: When you do you will find out! 
Actually they did slow it down quite a bit, using various maneuvers to both get data about performance at various yaw, roll, and angles of attack and to lose speed. The speed was down to only M=5 or so when I stopped watching the feed, so they had slowed it down considerably when it hit the water (it’s just that 50% of very very fast is still very fast).
Probably the only part of the Hyper X that is still intact is the big ol’ tungsten counterweight that formed the bulk of the craft forward of the engine. Everything else is probably scattered or trashed.