Read what the others said :-).
The original circuit diagram is here
This is the 555 based high voltage power supply section:
The HV via diode D1 is fed to the Geiger Muller tube to the right of this diagram.
There are 3 aspects of this circuit which a MC (microcontroller) would need to emulate to replicate it (as opposed to doing it differently).
An output high level on pin 3 starts a delay whose on-time is controlled by charging C3 using R6, D2, C3, and whose off-time is controlled by the decay of the voltage on C3 which is timed by R9 and C3. This delay could be completely replaced by software.
L1 is turned on by a high on 555_OUT turning on Q1. An MC would still need Q1.
Q1 remains on until the current through R8 + VR1 turns on Q3 thus ending the cycle. This sensing still needs to be done either by still using R8, Vr1, R1 & Q3 or by using an onboard comparator with level set either on chip or with a 2 resistor divider.
SO replacing the 555 based supply with a MC based design directly would eliminate very little of the hardware.
The 555 design "just works" whereas a MC based design needs to be failsafe against software crash. eg if Q1 is turned on an left on L1 may be destroyed.
A MC based design could do this more simply and probably just as well. Adjusting on time based on L1 current is not needed if Vcc is well controlled or can be varied open loop if Vcc varies widely and is known. Then -
- ALL that is needed to make HC is a microcontroller plus R5, Q1, L1, D1. C1.
Pulse Q1 on via R5 with some duty cycle and "HV appears". Adding a 2 resistor divider from HV would allow the MC to monitor and even control HV. The top resistor in the divider which drops most of the HV must be rated to max Vout. eg
Diagram may appear familiar :-). HV regulation could be eg 2 x resistors with onboard comparator or 2 x resistors to drive a resistor and a digital input pin or an HV zener and a resistor etc. OR eg a neon or HV zener could be used as a "constant voltage" load with maximum energy being designed to be within their dissipation capability.
Once HV is available, here is their complete Geiger Counter proper:
HV is applied at left. When the GM tube is ionised by a radioactive event (nature depends on tube used) a pulse of current turns Q2 on and produces an output pulse at GM_pulse. In the existing circuit this is the sole connection between the actual Geiger Counter (as described above) and the MC. The MC is effectively an "intelligent meter". There are enough spare pins to allow the above change to be made without excessive effort but it changes the whole nature of the MC program from that of a pulse processor to a real time controller (albeit a simple one).