The following one paragraph anecdotal explanation (with no equations) of why two clocks might be more stable in anti-resonance.

If we think of dynamic systems in terms of energy states, more energy is bad less energy is good. For example, a ball rolls to the bottom of a hill and stays there. The solution to the clock riddle falls out of this simple principle. When the clocks are any state but anti-resonance the frame that connects them would be caused to move and thus damp the clocks into the direction of anti-resonance (the ball rolls down the hill). When the clocks are in anti-resonance the motion of the pendulums cancel the motion in the connecting frame and thus the damping stops (the ball is at the bottom of the hill)

In a similar vein, my wife and I have the same problem. When we lie in bed at night, I have to wiggle my feet to get to sleep (commonly known as restless leg syndrome). If I keep my feet in anti-resonance, the bed moves very little and my wife can sleep, however, if my feet wiggle in phase (resonance) then my wife wakes up and does mean things to me (think of this as a damping force that causes the system to move into anti-resonance). I am sure if we spend enough time thinking about this solution we can base a whole philosophy (and maybe a new wave religion on it) but I will leave that to someone else and get back to work.