Given that this if for inside application the first thing I would state is to follow the manufacture's instructions. Depending on the stone weight you will have different install methods.
However... given no info from manufacturer:
I would not cement board over drywall. You would have to use longer screw of course but the fact that the screw isn't binding for the first half inch will put some serious strain on this connection. If you are dead set on doing it, glue it then screw it.
I would have have personally installed on drywall in dry areas. As long as this isn't a shower I don't see a performance difference between the drywall or concrete board unless the stone was extremely heavy
To prep for drywall install I would double up on the drywall screws, install wire mesh over the area, and then give it a small scratch coat over the mesh.
If you can give me specs on the stone I can get more specific or change my answer but nothing wrong with drywall as long as the area isn't wet (and given it isn't on a fireplace but then you shouldn't have drywall there already.)
Sheet of drywall is 48x96" or 4608 sq/in. The stones are each 108 sq/in. Meaning you will need 43 stones per sheet. The OP says the stones are 6.25 pounds, meaning the wall would be supporting about 260 lbs/sheet.
Each drywall screw can hold about 90 lbs of shear force without tearing drywall. An average sheet of drywall has 30-40 screws. Let's say 30 to give minimalist numbers. That means your sheet of drywall can probably hold 2700 lbs which is 10 times your stone weight.
My focus would be on reinforcing drywall and providing a good substrate - wire mesh. Adding concrete board to me only weakens the connection to the framing - basically you are saying that drywall can't handle the stone but it can handle the backerboard and stone. If you want concrete board then it should be on the wall - I would not use plywood as it does not bond to thinset as good as drywall or backer.
From a practical point of view... I have installed stone more than a few times on drywall. Your worries about the wall falling over are ludicrous. The weight isn't being pulled out. The weight is going straight down. 90% of the installation issues will be with the thinset and its bond. You are focusing on something that has little bearing on your finished product.