Have you read anything recently that estimated the # of Christian American Expats working abroad? Not missionaries, just people working but just so happen to be Christians?
No. I haven't. I don't think it can be readily figured out, either.
The American government apparently doesn't track how many people live overseas. This article cites estimates between 2.2 and 6.8 million. It also says that while Americans live in 100 countries, 66% of American expatriates live in 10 (including Mexico, Canada, Israel, the UK, France and Germany). This article, from the Association of Americans Resident Overseas, estimates 6.32 million, and breaks it down by country (with 2.59 million in the "Western Hemisphere" and another 1.6 million in Europe).
Estimating how many are Christians is even more difficult. We could theorize any population significantly large enough to be a randomized sample of the larger group would carry the same characteristics. If you randomly selected a few thousand people from America, theoretically they would have the same % Christian as the American population as a whole. But those who go overseas may not be a truly randomized sample. It's doubtful, to me. Very few Americans have passports; not many go abroad. Those who would be sent by a company to work overseas with a reasonable expectation of success in a cross-cultural environment may vary wildly from the general American population (unlike the Philippines).
So, I must confess to being a little stumped on this one. Some problems are intractable. If anyone has reference to studies, feel free to comment. But at the very worst, I would start with the idea that the expatriate population would probably have a makeup at least somewhat similar to the American population as a whole--quite a few Christians, some of them quite public in their faith.