They're both correct grammatically but the second sentence sounds more natural and would be how I'd say it.

Yes, your two questions make sense, but not in the context you describe. If you suspected that they were wrong, you'd say 'Do either of these sentences work?, as Paul suggests. You would only say "Do neither of these sentences work?" or "Don't either of these sentences work?" with a tone of surprise or disbelief. You would ask this if you thought that at least one of them was correct, but somebody had just told you - to your surprise - that they were both wrong. You would then seek confirmation by asking one of the above questions. You think that one or both of them is correct, and you are questioning the assertion that they are both wrong. I hope that makes sense!

Paul has the best options.

I don't think either sounds natural. I wouldn't use either of your options myself. I'd more likely phrase it as 'do either of these sentences work?' or 'which of these sentences (if any) is correct?'

I agree with other posters that both are grammatically correct, but the right answer is the second option, "Don't either of these sentences work?"