both do. the nuance is that hoặc is restricted to whatever you explicitly list (so it seems rude to use it in a question)
"She has either a cat or a dog" same meaning right? also seconding the question bout what the difference between hay and hoac
In English, in the vast majority of situations I can think of where this sentence would be used it would be a question - not a statement, therefore I believe it should end with a question mark "?".