I thought since he was born on a military base that this would not be a problem. The base is considered American soil.
Despite widespread popular belief, U.S. military installations abroad and U.S. diplomatic or consular facilities are not part of the United States. It is NOT sovereign soil. The land is still owned by the host country, but diplomatic missions are exempt from local law and in almost all respects treated as part of the territory of the home country. For example the host country may not enter the representing country's embassy without permission, but they still own it.
“...natural born citizen, or a citizen of the United States” -Article 3, Section 1, Clause 5. If he was born in Moscow in such circumstance that there is no paperwork necessary for him to be a citizen of the United States, he is a “natural born citizen...” They are just trying to plant the notion of cheating in the heads of people who have not read the Constitution, which is probably 99.xx percentage of the public school graduates in the last couple of generations.