The Internet is correct in this case. "Shuteye" means sleep. It is non-countable, so "some shuteye" would be the correct choice. Nap and snooze can both act as verbs or countable nouns, and so I would say, "I am going to have a snooze" or "I am going to have a nap". I hope that is clear.

"Nap" is the most simple and straightforward word. It means sleeping for a short time. "Snooze" means the same thing. It is a little bit informal. To me "snooze" carries the same idea of "doze," it may not be a deep sleep but a sort of half-awake, half-asleep in-between state. You might use "snooze" to minimize the idea in order to get permission. If I say to my wife "I'm going to take a nap" she might say "Oh, no, don't do that, we haven't got time." I might hope to get permission by saying "I'm just going to take a tiny little snooze." "Shuteye" is an informal, facetious (half-joking) way to saying a period of sleep. It could be long or short. It suggests something that is NOT a good long rest but better than nothing. For some reason the usual phrase is "catch some shuteye."