That is quite an unnatural way to say it. "Что вы думаете" doesn't work well with "or": usually it implies you should express your thoughts, not just pick an option (yes/no, Russian/American). "Что вы думаете?" can also be used separately, for example, when you are making an offer and then ask "So, what do you think?" (i.e. is it a bargain?).
Besides, as I already said, "Вы думаете, что Джей русский или американец" sounds as if you were reflecting on the fact that Jay is either Russian or American (that is, active thought process vs. having an opinion). Usually it would be "Как вы думаете, Джей русский или американец?" Remember, "как вы думаете" is a set expression, so it doesn't mean "how do you think" word-for-word.