This article is written for the benefit of the hundreds of foreign nationals who seek employment in American private schools each year. Be aware that most American private schools will hire a foreign national only if he or she has a Green Card. The Green Card or Permanent Resident Card (also known as Form I-551) gives its holder permission to work in the U.S. Without it or American citizenship you cannot work in the U.S. So the answer to the question "Can foreign nationals get teaching jobs in U.S. private schools?" is "No!" with only a few minor exceptions.
Why won't a school sponsor your application for a Green Card? Because it is too much of gamble. The school goes through all the hassle and expense of securing a Green Card for you only to have you decide a few years later that you want to work in another school. It's just not worth it.
A few private schools will go to the bother of securing a Permanent Resident Card for a key staff member such as a head of school. But those instances are few and far between. The situation is very much a 'catch 22' as the American Embassy in your home country will tell you that you need a job before you can apply for a work visa. On the other hand the school will tell you that it needs you to have a green card before it will consider you for employment.
The very few exceptions to this general rule are specialist teachers such as those who teach Chinese (Mandarin) language. The NAIS China Connection Program
is an example.