I presume you mean when inflight? No, there is no 'critical engine' on any of the jets that I've flown.

Could be an issue with some engines having different systems than others but that doesn't come into play until you have more than two engines. i.e. the DC8 had engine-driven hydraulic pumps only on the inbound engines.

To expand on the single-engine taxi answer above... I mentioned that on the 737 we always start the right engine first due to the configuration of the pneumatic system. I failed to mention that by starting the right engine first you can then maintain heating/cooling during the start process on the left engine, by having the right engine provide bleed air to the right AC pack, while the APU bleed air operates the starter for the left engine. If you started the left engine first you'd have to have both AC packs off while starting the right engine.

Additionally, when we shut down one engine during taxi-in it is the right engine that is shut down first leaving just the left engine running. I don't know why it's always that right engine first but there's no longer any benefit in running the right engine so, perhaps, it is to balance out the operating time between engines?

I think, but am not sure, that the Airbus is the opposite.