It's interesting how much rabbinic attitudes can shift over time. Back in the Second Temple era (and even up to Nachmanides), a Jew going to a Gentile court was completely anathema. We even see this attitude in Paul's writings, though Paul changes the classification from Jew/Gentile to Believer/Non-Believer. The underlying rationale seems to be that an erosion of communal autonomy even in the realm of law will eventually lead to communal disintegration.
So here's selections on this topic that I read today from Menachem Elon's "Jewish Law":
pg. 13 "At about the time that the Temple was destroyed, when the Roman authorities restricted Jewish legal autonomy for a short period, we are told:
'R. Tarfon said: 'In all place that you find agoriot (non-Jewish courts), even though the substance of their law is the same as Jewish law, you are not permitted to resort to them, for it is written: 'And these are the laws that you shall set before them.' Before them [the Jewish people] but not before the gentiles.'
....this pronouncement established one of the sturdiest bulwarks protecting the continuous existence and development of the Jewish court in all the periods of the exile. Resort to a non-Jewish court was compared to the denial of the existence of God and His Torah and to the transgression of the commandment against profanation of God's name. 'Anyone who litigates before non-Jewish judges or in their courts, even though their law is the same as Jewish law, is an evildoer, as if he has reviled, blasphemed, and raised his hand against the Torah of Moses, our Teacher....' [footnote 39: Maimonides, Mishneh Torah...Sanhedrin 26:7...]"
pg. 15 "...the prohibition against the use of non-Jewish courts was more strict than the typical legal rules relating to civil matters, since the use of such courts had the effect of undermining the essence of Jewish autonomy."
pg. 16 "...it was Nahmanides' opinion that the case must always be tried before a Jewish court, and the only legitimate function of the non-Jewish authorities in a dispute between Jews is to enforce the judgement rendered by the Jewish court. However, the distinction made by Nahmanides [that a Gentile court could enforce a ruling made by a Jewish court] was not accepted, and the rule as finally established was that one must first seek relief before the Jewish court; and if the defendant refuses to appear, the claimant, with permission of the Jewish court, may bring his claim against the defendant in the non-Jewish court in order 'to prevent the strong and powerful from evading the law.'"