My wife wants a new tile bathroom floor. My subfloor is 1x6's laid at a 45deg. angle across the floor joists with 4x8 sheets of particle board on top of that. I want to take up the particle board and put down plywood on top of the 1x6's, but I'm not sure what thickness I should use. I understand that tile needs a strong support subfloor and hardibacker under the tile. So, I would like to know the suggested thickness of the plywood and hardibacker that I should use.

Definitely remove the particleboard. If the 1x6's are in good shape and you have enough height. Install 3/4" ply perpindicular to the joists.
By the way; what do your joists look like?
What size boards?
Spacing?
Length of span?
Type of wood?
Sound condition?

1/4" Hardi is all you need for the floor. Or you can use Schluter Ditra. It's only 1/8" thick.

The floor joists are 2x10 on 16" centers with some cross-bracing between joists. The area I would like to tile is 8x21. The house is 28ft wide using 14ft joists so the span of each joist is 14ft. So the floor would cover all of one span plus 7ft of the other. I suspect the joists are Southern Pine, but not really sure. They look to be in good condition.

Your deflection on that floor should be fine for ceramic. It falls short for natural stone.
If you have SYP joists; and those 1x6's are still in great shape. I would assume minimum 1/2" but 3/4" ply is always better.
Run the ply perpindicular to the joists. screwing no deeper than the thickness of the two layers. Don't use screws long enough that they will penetrate the joists.
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