I tried pushing on both sides of the rear to compare. They both come back up past the equilibrium point a little bit, then go down to the equilibrium point. So the shocks are working decently well, and I'm thinking the right-rear shock probably isn't the cause of the noise.
That is a good point about the probably coming from sway bar type bushings. Given that the clunk comes on flat rough pavement it seems like it is caused by a rearward force rather than a vertical force. The Haynes manual for my car doesn't list a sway bar in particular, but it has a strut rod that runs horizonal lengthwise down the car to restrain the wheel in the front/rear direction. Seems like the bushing at one end of that rod may be dead. Is that something that I should be able to visually identify?
Autozone sells a sway bar repair kit for $53. Is this probably what Haynes is calling the strut rod? It's hard to tell from the pictures, but if they're not the same I have no clue where the Autozone part would be installed.