Often, Jewish organizations have as a contract clause that employees agree to take issues to beit din and not (or at least, before) secular courts. But if the employee is not Jewish, is that employee's first recourse still required to be beit din? I'm not sure if this is a question of law (the validity of the religious court as a viable option recognized by the US legal system and the contract's clause still being binding) or one of halacha (does the beit din have authority over a non-Jewish party).
Journal of the Beth Din of America 1:1, page 32:
-3. Non- Jews: Tashbetz assumes that, technically speaking, the prohibition against litigating in secular court would apply even in the context of a non-Jewish adversary. However, one may assume that a non-Jew will not willingly appear before a beit din, and accordingly one may bring the non-Jew before a secular court without permission from beit din.
My understanding is that sometimes a non-Jewish party will agree to use a beit din -- compared to the regular courts, it can be faster, cheaper, more confidential, and/or it's more likely that the Jewish party will be compliant. (Rabbi Reiss tells the story of a medical malpractice claim -- the insurance company far preferred a confidential panel of three experts over a long, expensive, public trial by jury. If the non-Jew is okay with a beis din, then the Jew has to use it!)
As I understand it, though I am not a lawyer, an employment clause can say "issues shall be settled by Arbitration Panel XYZ", and if people sign that, they would have to go (at least initially) to Arbitration Panel XYZ. Generally a beit din is recognized by the government as an arbitration panel if both parties agree to use it.