The negative supply serves as a sort of passive current compensation..
You'll see that, if you sum up all different currents, like through the zener, and both transistors, and compare that value to the current through that strange dummy resistor to negative supply.
Anyhow, this is not needed, and gives no big advantage, as the circuit is an exact copy from the datasheet.
Therefore, you can run it from any single positive supply, like TiN already wrote.
Again, before attempting to make precision measurements, proper thermal and electromagnetic shielding is required.. this board is sitting inside a double shield in the 3458A..
And your measurements already showed many spikes from external sources (SMPSU?, AC mains line switching?) and maybe popcorn noise or influence from air draught (these longer fluctuations).
Frank