A couple of thoughts:
When a person makes a claim about a physical event in the physical world, it is up to the person making the claim to present evidence, and not up to the person being asked to accept the claim to “have evidence for rejecting the claim.” So, when a person makes a claim of supernatural events in the world, those listening (be they Christian or non-Christian) must evaluate the evidence of the person making the claim and then accept or reject it on that basis. In a sense, then, (unless I’m misunderstanding you), you’ve got it backward: Christians don’t need to provide evidence that rejects claims, in the same way that no one needs to provide evidence that someone else’s claim doesn’t hold up to scrutiny.