I still scratch my head when I hear this objection: "If God possibly exists, then He also possibly does not exist."
The first step in refuting this (sadly) oft-repeated criticism of God's necessity is to explain that the opposite of "possible" is not "possibly not," but "not possible" (impossible).
What the objector is confusing for possibility is contingency. A few modal definitions should suffice to clear the air:
necessity: existence in all possible worlds
contingency: existence in at least one, but not all possible worlds
impossibility: existence in no possible worlds
Obviously, "necessity" entails possibility, whereas the latter does not entail contingency.