Low noise amplification - Op Amps vs Transistor based amplifiers
Specifically for EEGs there is no real way around FET input instrumentation amplifiers. That is the only strategy to get good CMRR at high gain (in-amps actually get better CMRR performance at higher gain, which is exactly what you need). I don't see how class A has anything to do with it, that is not a specific technology but just a way to describe a broad range of audio application amplifiers. There are class A amplifiers with very high noise and with very low noise.

You will be limited by physical noise in any case, no way around that. So make sure you have as good a signal path going in as possible, shield the amplifier and use coax braided shield wires to the electrodes. EEGs are especially sensitive to electrostatic fields which usually causes irreconcilable issues with mains hum. Therefore you need to digitally filter that out - there is no better filter than a digital one and you need all the precision you can get.