What do I use to level out with between green board and hardy backer board?
You can use thinset mortar to float the gap. If the difference in surfaces is significant then you will need to float it out on the low side quite a bit. 

It does not need to be to pretty but it should be relatively smooth. Use fiberglass mesh tape made for hardy backer on the seams to add strength; make sure the screw heads are counter sunk properly.  

For 1/6 inch I am not sure it is necessary.