Non-Jewish Courts

What is the halacha concerning joining and participating with non-Jewish courts?

It is permissible in Chutz laAretz – dina demalkhutha dina. See TB Nedharim 28a, Gittin 10b etc. Keep in mind too that gentiles are required to establish a court system: see MT, Hilkhoth Melakhim, Chapter 9. The exception would be when two Jews have a legal wrangle. They are both hayavim to turn to a proper beth din. If one refuses, he can receive permission from the beth din to turn to a gentile court.