Some brands have released protectors for the 5S and it's updated home button and such. For some existing brands, iPhone 5 protectors already work. For others, they're entirely incompatible with the iPhone 5S.
You'll just have to do some research. Off the top of my head I know that Zagg's protectors are 5s-compatible.
Sorry perhaps I should clarify, Zagg for instance has a very wide home button opening, Spigen has a very close fitting opening, Power support has a wider home button opening that Spigens. I have never seen the size of the home button on the 5S so I don't know if its the same size as the home button as on the 5 but if the screen protectors home button opening is much wider than the 5 home button than I don't see a compatibility issue.
I believe the Apple folks said "yes, but..." with the caveat being that the opening for the camera is different because of the dual-flash update in the 5S. A 5 case may interfere with the 5S flash.
It seems like the answer would be no, because the home button opening would have to be bigger (because of the ring). I get my screen protectors cheap on eBay, but usually the home button is pretty tight and close to the actual home button size.