If a US State passed a law regarding the eligibility of Presidents, could it be enforced?

Suppose (the citizens of) a state were tired of government shutdowns and they or their legislature passed a law prohibiting members of Congress and Presidents involved in a governmental shutdown from being elected again. Obviously, a Democrat might ignore such a law if it were passed in Wyoming and an especially popular president might look at the electoral math and run anyway. I'm aware that many states attempted to pass laws regarding birth certificates after Obama won in 2008. It doesn't look like these measures got far.

• Could a State deny a candidate a spot on the ballot?
• Could a State deny its electors to such a presidential candidate even if they won a write-in campaign?

Obviously, a well-funded candidate would challenge these measures in court - the question is: who would win?