Anode driver for Nixie clock
What might be the reason why the single MPSA42 configuration would not switch ON?


Actually the transistor switches on, but the voltage you can reach on the emitter is smaller that the voltage at the base. This configuration is called "emitter follower". 

An NPN transistor will "conduct" if the base-emitter junction is forward biased. Therefore it must be \$V_b-V_e&gt;V_f\$, where \$V_f\$ is the base-emitter junction forward voltage (between 0.6V and 0.7V).

Since your Arduino output is 5V, the maximum voltage at the emitter will be \$V_e=V_b - V_f= 4.3\ V\$ (assuming a negligible \$I_b\$).


  Why would the other design add the MPSA92 and whats the logic behind
  using the MPSA42 + MPSA92 combination?


This circuit has a big problem: you must put a resistor in series to the base MPSA92, as follows (othewise the both BJT will likely be destroyed). OR you can change the MPSA92 with an MPSA42 (NPN).







simulate this circuit &ndash; Schematic created using CircuitLab

Assuming you use the circuit on the left:
The first transistor (MPSA42) acts as a level translator. When it's OFF, the collector voltage is pulled high (to +HV). When it's ON, the collector voltage is about 0 V. 

The second transistor, in turn, (MPSA92) is a PNP BJT. It will be OFF when the collector of the first BJT is at high voltage (Because its base-emitter junction is 0V), and it will turn ON, when the collector of the first BJT is at 0 V (because there will be a current being injected into the base of the MPSA92).

If you use the circuit on the right: the first transistor is a level translator, the second one in an emitter follower. The difference with with your first single-transistor circuit, is that now the base voltage can go high. At this purpose, knowing the anode current you should properly calculate the resistor values (watch-out for power dissipation too!)