There was quite a bit of discussion about using screen protectors with the iPhone 5 Lifeproof Nuud case on that thread, so if you want a lot of detail you can go over and read all 30 or 40 pages or whatever that thread wound up being.
However, to save you some time, here is the general consensus:
Lifeproof definitely states that the case is designed to be used without a regular screen protector on the phone. The tolerances on the case are *very* tight to allow the case to be waterproof and even adding a fraction of a millimeter in thickness could put enough strain on the seal where the back of the case connects to the front of the case to cause a leak. Additionally, as far as antiglare screen protectors go, there was some concern that the texture used to create the antiglare effect would cause issues with the seal where the case meets the phone screen and could cause leaks there.
Now, that said, if you just like the lifeproof case for the way it looks or for the drop protection and don't plan to actually use your phone underwater, then you could probably get away with leaving a screen protector on. I would imagine that it would seal well enough to keep out a splash of water here or there or protect your phone in the rain. But if you're going to go swimming with your phone or want to take pictures underwater, then I would suggest not using a screen protector and using the case as lifeproof intended to be sure that there will not be any leaks.
As far as the antiglare screen protector goes, assuming that it's just a standard plastic screen protector, then you could cut it down to fit over the exposed part of the screen the way the screen protector that comes with the case works. Obviously this won't work for glass screen protectors and it's not a good solution if you don't plan to leave the lifeproof case on full time.