Can I use cement backer board as a subfloor?
No, Hardie cement board is not meant to be structural, and it will not work in place of a subfloor.  The board is very strong in compression and very stable so that the tiles applied to it do not crack, but it has very little shear strength.

There are treatments you can put on wood to make it less attractive to insects.  I would recommend approaching the problem from that direction and use traditional, proven materials for the subfloor.