I took my Civic to my mechanic on Tuesday, June 27th, for some new
tires.  While I was there picking it up, I mentioned that I was unable
to get the cover off of my air filter.  He volunteered to give it a
shot and told me it shouldn't take him long at all.  Not to my
surprise, time quickly passed and a half-an-hour later, he was finally
making progress on the cover.

The problem appears to be exactly what mboscott suggested: the heat
from the engine has weakened the plastic around the bolts that hold
the screws in place.  As a result, when you attempt to turn the screw,
the bolt isn't held stationary by the plastic as was intended.
Instead, the bolt spins, and in my case, this spinning caused the
plastic surrounding the bolt to split in places.

The solution is simple: find a new way to hold the bolt while
unscrewing the screw.  My mechanic and I just broke off the plastic
around the screws enough that we could fit a wrench in to hold the
base.  Of course, this won't fix the problem permanently, and will
only work if some of the screws are not suffering from this condition
(in my case, the screw in the center and the two screws on the side of
the box opposite the engine were still usable, and are now the only
screws that hold the lid down on my air filter).  The only permanent
solution I can think of would be to replace the stock air filter box
completely with a box made of more durable plastic.

Hope this helps, good luck!