Synchronous demodulation with single supply op-amps
The problem is the square wave from the sensor is centered around zero, and the switch rectifies the signal back to DC. If the op amp's negative rail is tied to ground, the range is 0V to +Vcc which means that half of the square wave. Actually now that I think about it, this should be able to work, because the amplifier is essentially switching between a gain of 1 and -1 its kind of like applying an absolute value to the signal (as long as the signal is synchronous with the chopping signal, sometimes delays can create error in these types of circuits, and not like an absolute value in the sense that it can go negative in the circuit above). If I have time, I'll run some calculations to see if this holds true, it's been a few years since I have messed with a circuit like this. 

However, this would affect the range of the sensor on the other end, which I would imagine that it would be restricted to above 0 as well.

The real power behind these circuits is to be able to restrict the noise to a certain bandwidth. Back in the day 1/f noise was prevalent on most amps, so circuits like this were used to modulate the signal out of that frequency domain, and a lock in amplifier like the one above to bring the chopped frequency back to DC. Another good tool is stick a bandpass filter before the lock in amplifier to restrict the bandwidth of noise even further.