What are my vapor-barrier options under Hardie siding?
You have asked several questions at once so a little difficult to give you the answers you want. 
First, Hardy backer board is a newer, lighter,  substitute for concrete board and usually used in bathrooms and under tile installations. I have never seen it used on exterior walls as a sheathing. 

Second, the huberZip system is from Advantec. I love Advantec sheathing and subflooring. I'm sure it does what it says, but it is pricey compared to OSB sheathing and tyvec. Selecting a product because of price is not what we do here. An OSB, or any T&amp;G exterior sheathing with properly installed and taped Tyvec or a house brand wrap also works very well. 

Third, a true vapor barrier is installed on the heated side of the wall studs, not under or over the exterior sheathing. the wall cavities and insulation must have a a way to breath. They will breath and excessive moisture should escape through the exterior siding. Tyvec is not a vapor barrier, it is an air penetration barrier, just like the new Zip system. they both stop liquid water infiltration, but water vapor is a different story, it must be able to pass through. On wet, foggy or very humid days, humidity gets into the wall cavity, the vapor barrier keeps it out of the living space. When the weather conditions are better, the moisture migrates out of the wall cavity through the siding, thus drying the wall.

I hope this has answered some of your questions.  Good luck on your project.