no and no.
Hardi, Durock and greenboard are all different in several ways.
If you use Hardi board, in a shower, the recommended installation is to put a plastic sheet between it and the studs. The plastic is then a "vapor barrier" preventing moisture from migrating into the studs, the wall cavity and the rest of the house. Many people have not had vapor barriers installed in their tub-shower, and have had no problems. The degree of risk is not determinable in advance. Construction methods cannot rely on indescribably perfect installation which also depends on a little dose of dumb luck to succeed. It is not a recommended practice to follow. Hardi looks and feels like it's waterproof, but it isn't. Check their web site. Hardi is not waterproof; it can let moisture migrate, it can hold water.
Why a vapor barrier? Nobody should ever rely on grout alone to block water, since grout is not waterproof itself (it can hold a lot of water) and it may be insufficiently applied in one or two places; although it can look continuous on the surface, it may cover big air spaces behind or between tiles that grab water and hold it. Pure water doesn't enable mold to grow, but soap and construction materials are food for mold. Once conditions are ripe for mold to grow, it can grow fast and give off mycotoxin gases and liquids, which might remain largely trapped behind the tiles. This explains why some showers "smell" and why powerful smells are released when showers get demolished. These mycotoxins are true excrement designed to eliminate molds' very own fecal matter; they are so powerful that they even act as chemical warfare among microorganisms, preventing other microorganisms from growing in their vicinity. That is how insulin was discovered, and that property is what makes insulin work for us. Most other molds give off excrement that hurts us; it weakens our immune system by forcing it to work very hard all the time.
The plastic sheet directs the flow of moisture down to the inside of the tub's tiling flange. This is good. Now, invisible water vapor has a direction to follow. Either it evaporates back out through the grout, or it slips down (percolates down) and collects into drips that go down into the tub or evaporate at that bottom edge where the collected water would have dripped off if it had been big enough to drop. Although a plastic sheet gets perforated when nailed to studs, this is not deemed to be a serious problem; people have not reported that studs have suffered at nail holes nor harboured mold there.
If you use Kerdi, the continuous layer of vaporproofing is now Kerdi, and not the plastic sheet. Kerdi needs support, so it goes on top of your wall surface, not underneath. Being waterproof, Kerdi ensures that the wall surface won't get wet; this explains why you can install Kerdi on top of any drywall. You don't need Hardi. You can still use Hardi, or Durock, but it is not required.
Squeaky floor:
you have a problem that Ditra may not fix. Squeaks are movement. Movement is bad. Study this first before choosing the under-tile membrane to use.
David