What makes worms' eggs float or sink?
Okay, I found the last part of your question the most interesting and can't give a full answer to that, but I just noticed I can answer the rest so here goes.

How it works (the simple version)
It works on the principles of buoyancy. The eggs and oocysts are less dense (lighter) than the fecal flotation solution and, consequently, they float on the very surface of it. The feces themselves are more dense (heavier) than the fecal float solution and, as a result, they sink to the bottom of the solution, out of the way. Commonly used flotation solutions include sodium nitrate, Sheather's sugar solution, zinc sulfate solution, sodium chloride solution and potassium iodide solution.

The eggs of some important parasites (for example. Schistosoma species and Fasciola hepatica) are more readily diagnosed by fecal sedimentation (sediment) tests because their eggs tend to sink in flotation media.

What makes it so important?
Some parasite eggs float and some don't. Faecal flotation is a simple, rapid and helpful diagnostic that narrows down the agent causing the clinical signs in the animal. 

How it's done
Use a "Fecalyser", a commonly used test kit for faecal floats. The best stool samples to get are those that are fresh - straight out of the animal within the last 30 minutes where possible. Detailed walkthrough of the protocol here and video tutorial here.

Why is there a difference in the buoyancy of fertilised versus unfertilised eggs?
No idea, sorry.  Similar effects are seen in some insect eggs - Stomoxys eggs which sink mostly hatch, while floaters don't hatch. The fact that Anopheles eggs have floats and Culex lay egg rafts implies to me that their viable eggs would otherwise tend to sink too (this could be to do with oxygenation levels being higher at the surface, but I'm speculating here). During embryonation, a serosal cuticle forms which makes the eggs highly desiccation-resistant; non-viable eggs dry out. However, according to some sources online it sounds like for parasite eggs it's the unfertilised eggs which sink, not the fertilised ones.