You do need a constant current source, but the best way to do that is to use the Zener voltage to regulate the current in the current source. So if the zener voltage is stable, the current source is stable.
All you need is a circuit like this:

Now the zener can be either a zener alone, or a slightly higher voltage zener in series with a forward biased diode junction with the opposite coefficient.
Start off by assuming the voltage across D1 will be constant - that is the aim after all. The voltage on the opamp output will be a constant voltage (about double the zener voltage) and so the voltage across R1 will be constant. Since the voltage across R1 is constant, the current through R1 (ie the current through the zener ) is constant.
The secret to stability is the zener temperature coefficient changes with current. Typically as the current increases, the temperature coefficient decreases. So if you start with a zener that is very close to zero temperature coefficient, then by testing the coefficient, and then adjusting the current, you can find the current with the minimum coefficient. Increase the current and the coefficient goes negative. Reduce the current and the coefficient goes positive. Somewhere in between, it is very close to zero. That is why in Dave's reference box, they included the calibration number. When the current is adjusted to give that voltage from the zener - it is at the point that the manufacturer measured the lowest coefficient. It is a number that only works with the current zener, and if the zener had to be replaced, the whole unit would probably have to go back to the manufacturer to determine the best operating point for the new zener.
In the circuit above, you could vary R1 slightly to set the current.
The resistor coefficients come into it, but if you calibrate for zero coefficient using this circuit with all parts in place, then you can pick the zener current that gives lowest net coefficient. So if the resistor and opamp coefficients end up giving a slight positive coefficient to the current source, then you pick a current where the zener has a slight negative coefficient to compensate.
Put the whole circuit in an oven, and the thermal stability improves a couple of orders of magnitude, and you have the big advantage in that you only have to find the best coefficient for one single temperature operating point.
Also the other key to a zener reference is the voltage at which the zener is most stable is not important - what is important is that the voltage is extremely stable. You then calibrate elsewhere.
Can you make a stable reference with cheap standard zeners? You could be lucky, but the real reference zeners are often pre-aged and often have a construction that minimizes the effects of the surface conditions on the silicon die on the zener voltage. Many of the best zener references use buried zener diodes. If you did use a cheap zener you selected because it had a zero coefficient between 5 and 10mA, I would probably keep it running for a few years, and then recalibrate it for zero temperature coefficient. At that point, it will be much more stable then when you built the reference.
Richard.