As I understand the issue, the VA routinely puts these folks in the NCIBC now and the amendment is intended to stop that practice.
From Huffington: "The number of veterans directly affected by the VA's policy doesn't appear to very large. Only 185 out of some 127,000 veterans added to the gun-check registry since 1998 have sought to have their names taken off, according to data that the VA shared with lawmakers during a hearing last June."
What's large IMO is the 127K vets who are already being treated like dangerous criminals. If a VA shrink determines a patient is a danger, I have no problem with dangerous vets getting the same treatment as anyone else. But this VA current policy is a blanket overreach that must make the gun grabbers and the anti-military folks very happy. Maybe they figure vets are more likely to resist enemies of the Constitution.