There's some confusion here about Hypersync because it's used to mean two different things.
At it's simplest level, HS fine-tunes the timing to squeeze the flash into the shortest possible shutter-fully-open duration. This is limited by the camera, and that's where the 1/250, 1/320 numbers come from. Faster than that, the curtains are never both open so you can't have an instantaneous flash exposure of the whole frame no matter how you tweak the timing. This is with Speedlights, which have very short duration.
However, at faster speeds, operating much more like HSS, it relies on either a long flash or a long flash tail to expose the frame even though there is never that 'one moment.' This is typically the case with studio strobes. It really shouldn't be called the same thing as the first one.
The OP was asking about flashguns, so the first scenario is the most useful for him.