After disconnecting and reconnecting everything attached to the receiver and giving the area a thorough clean, I was still encountering the problem - my A speakers were still clicking on and off at random intervals. I then happened to notice an error on the front of the receiver that said "Phones In" when I tried to go to the receiver's home menu, but nothing was actually plugged in. This was my ah-ha moment. Sure enough, I plugged a set of phones and then removed them, and my A's were available again. I used a q-tip and a bit of rubbing alcohol to gently clean the port and haven't run into the problem in the 2-3 hours that I've had it on since then.
I'm not sure what actually caused the problem in the first place, but hopefully it was just a bit of dirt or or something in the port. If it happens again, I at least know the problem and it should be easy enough to fix.