You don't need any kind of special server that I am aware of. Unless the are referring to the Flash Communication Server. You may want to ask and see. Otherwise it just seems like some sort of ploy to add "features" to their hosting services. An example of this: My brother got new computer speakers for Christmas. As listed features they had...
"Windows XP Compatible"
...and other nonesense.
Just throwing out words that consumers will latch onto.
You site chugs in flash because you have a lot of complicated large vectors moving across the screen. The reason quicktime plays faster is because it prerenders each frame as a BMP and then just flips through the images. Flash renders as it plays...doing lots and lots of calculations. Quicktime just shows a string of images.
The playback speed of your movie is dependent mainly on the end-user's machine and not the server it's hosted on.
Hope that helps...and if I'm off on anything someone please correct me.