Just an idea - again, with the caveat that you might want to control the "piggy back" boards boost converter - pretty sure all boost converters have an enable line, and you should power up logic before VEE on most STN LCD's anyway.
Most converters use a ~1.25V reference on the FB pin. I assume the trimpot used to adjust this is one of the resistors in the feedback path. As you have access to the "old" negative VEE, then a resistor from this to the FB pin instead of a resistor to GND should give you want you need. If the values of the resistors in the voltage divider on the feedback are the same then, a -17.5V from the negative VEEE should prodice ~+20V out of the boost converter. Similarly a -14 input should produce a ~+16.5V on the boosts output.
So the adjustment range on the original board was:
-17.5 to -14VDC
Making the divider the /2 by making both resistors say, 20k, then the output is (-1*VEE)+(2*Vref) = -VEE+2.5V. Giving a range of 16.5 to 20V. You could adjust the values of the resistive divider to shift this slightly higher to give your +17 to +21V.
Again though, the main caveat, which I shall now put in bold:
Normally the feedback pin has a resistor to GND, which is generally a pretty stable and solid reference. Referencing this to an adjustable source means the output of the boost converter is dependant on this, and so it should be reliable and stable before the boost converter is started.As most negative supplies when they start up - just like boost converters - start with an output at 0V and then go negative from there. The effect of this is that if you power up the boost before the negative supply, the boost converter will output a voltage of ~2.5V (feedback pin = 1.25V, both R1 and R2 = 20k, so output is 2*1.25V = 2.5V. And wil increase as VEE goes negative. So it shouldn't blow anything up.
I would still add some kind of delay before powering up the boost converter.
So, yeah, you might be able to do this with a single resistor.