Basic schematic help
R1 and D1 will not limit the voltage to 20V. The resistor would need to be in series with the +24V supply rail for this to happen.
In any case, why do you want to limit the voltage to 20V if you are limiting the current through the resistors? The current limiting circuit will adjust the voltage across the resistors to the same value either way.  

ZA and RA just limit the gate voltage to 5V (or not - I didn't look at the values, Steven is right there), in case you want to interface with a microcontroller according to the Instructable. The only effect this will have on the current limiting is setting an upper limit to the MOSFET turn on (which may be a good thing, but not necessary)
The circuit will limit the current through the LEDs just fine like this.   

To show what happens with the circuit over a range of supply voltages, here is a quick circuit I created, without the unnecessary bits:



Simulation:



I set the current for around 20mA using 0.7V / 0.02A = 35&Omega;.
The X axis is V+ (the supply voltage) being swept from 10V to 30V. The green trace is the voltage across the 3 LEDs, and the blue trace is the current through them.
You can see how once the voltage rises high above the LEDs combined Vf, any further increase has very little effect on the voltage across them or current through them (less than 1mA variance over 20V)