Does it matter the order of amplification and rms squaring?
I read your question as meaning that if instead of feeding a real signal into the amplifier you fed into the input a DC level representative of its RMS value, will the output DC level be the RMS of the amplified signal.

If this interpretation is correct then theoretically, on a perfect amplifier with no DC-offsets to worry about and providing the amplifier is DC coupled in its stages then the output DC voltage would be OK but there are a lot of things that could make this wrong.

For instance if the real signal were somewhat attenuated by frequency shaping circuits (even just a regular 3dB roll-off as seen in op-amp circuits) then the real signal at the output may be significantly less than what you'd expect. This of course means the DC version would give a falsely high representation of the RMS.

This is just one example.