Switching negative voltage
That's just one circuit, which happens to produce a voltage inversely proportional to the resistance.  There are many other possibilities, some of which don't require a voltage outside the output voltage range.

If this is going into a microcontroller, and in just about any application it would nowadays, the output doesn't have to be proportional to anything in particular.  In a micro you can easily do the math or do the lookup to convert from whatever A/D reading you get to linear pressure.

The simplest option is just a resistor divider, with the other resistor in the middle of the range you care about coming out of the sensor.  One side advantage of this is that the only thing you have to calibrate is the other resistor.  The actual supply voltage cancels out.  You will have highest resolution at that resistance, then lower at both ends of the range.  This method actually measures from 0 to infinite resistance, but again, the resolution goes down at the ends of the range.

With today's A/Ds, I'd probably use the divider method and make up any loss of resolution with a higher resolution A/D.  You can now get 12 bit A/Ds routinely in low cost micros.  That's probably enough, but if not, a external delta-sigma A/D could certainly do it.