DY: Note the Chazon Ish’s language in referring to Jews who serve as judges on non-Jewish courts as “Jewish judges who have traded away a Torah system for a worthless, vain system.”
Secondly, and more importantly, a Jew who accepts a position as a judge in a non-Jewish court, especially in an area where Jews live, will unfortunately face many Jewish litigants, whether as plaintiff, defendant or both in the same case. And the Jewish judge is then halachicly prohibited from adjudicating that case involving Jews by applying non-Jewish laws. But his position as a secular judge requires him to judge the case using non-Jewish laws and non-Jewish standards of judging. Thus he will inevitably be forced to violate halacha when faced with Jewish litigants.
Virtually none of the Jewish judges on the non-Jewish bench recuses himself from all cases that he is assigned that are involving Jews. If a judge were to recuse himself so many times it would be obvious to the public and press that he is doing so for that reason. It doesn’t happen. And very frequently one of the Jewish parties filed a case in arkaos, contrary to halacha, and shlepped the other Jewish party to arkaos unwillingly. Then they face the Jewish judge who will unhalachicly adjudicate the case via non-Jewish laws and standards, against the will of one of the parties – because the other party refuses to withdraw the case and go to beis din. (Note, though, that even if both Jewish litigants were willing to have the case adjudicated in arkaos rather than beis din, it is still prohibited to do so.)