1 1/4" rock-on screws wont countersink. I'm trying to fasten 3/8" hardibacker to 3/4" particle board for a countertop finish with epoxy, so I need a flat finish surface. Is there another screw to use or different threads to use to get them to countersink?

There is a difference between "rock on" and "backer on" screws. I believe you need to use the latter to countersink with Hardiebacker.

Why not drill the hole, use a countersink tool to form the angle and then drive the screw flush?

Assuming that "rock on" screws are the type with the barb under the head, you may need to use longer screws. Particle board does not have much gripping power and short screws will eat out the hole before pulling the head down below the surface.