Possible, but unlikely. For example, if we have a conventional (i.e. not bomb-pumped) X-ray laser, tightly focused enough and fired from a ground-based platform, then if we miss, we'd basically just be firing a huge burst of X-rays into space. The atmospheric results would likely be anomolous (that kind of ionization doesn't just happen for the hell of it), and the timing would be suspicious, but although a lot of people would guess that we had tried and failed to knock the missile down with an X-ray laser, no one would really have proof.
The problem, of course, is that for military purposes, suspicion is almost as good as proof. In particular, the kind of facilities (particular power generation) necessary to fire one of these devices more or less requires a dedicated, land-based facility (probably based in a friendly country like Japan, due to the need for access to the boost phase.) You could never fit such a beast onto a sub or an aircraft. Once the NKs know that the US has deployed such lasers, they're an obvious espionage or sabotage target -- again, we've basically blown our cover.
A more conventional system, such as the Patriot, will rely on kinetic-kill missiles. These can be deployed by mobile units, but are very easy to track on radar, and the kinetic-kill (or, more likely,
the kinetic failure-to-kiill) will be obvious to any observers.
