Could spy satellites use laser guide stars (for adaptive optics)? Are sodium lasers useful for Earth observing space telescopes/spy satellites?
The guide star laser is used to adapt the optics at a specific time and location, for specific atmospheric conditions. It is not, as your question implies, used to calibrate the optics for long-term use.

Therefor, if order to calibrate the satellite with a laser fired from Earth, the reconnaissance target would have to provide the laser. This is useless for non-cooperative targets, which I would imagine make up the bulk of the reconnaissance targets.

I suppose that one could fire a laser from the satellite itself, but this would have the obvious disadvantage of making it even more obvious when a spy satellite was overhead.

In any case, the guide star image does not penetrate the entire atmosphere, I believe that it produces an image just a few tens of kilometers up. This is useful because it is the thicker, denser and more turbulent lower air of which is the most concern to optics. Firing the laser from above would thus see the laser energy absorbed in the thinner, stiller, more predictable upper atmosphere, thus it would have must less advantage.

One final reason not to use a guide star laser on a reconnaissance satellite would be power budget. These satellites are as small as possible, with as small a heat and reflectivity index as possible. The added power source, be it RTG, conventional batteries, or solar, would likely increase the reflectivity of the satellite.