Circumcision that wasn't done for the sake of the mitzvah (or for God) was considered invalid. Of course, whether one needs that degree of intent is up for debate. Note also that non-Jew who happens to be circumcised isn't fulfilling the mitzvah of brit milah. Also, a new mishnah: medical or other health treatments by professionals who are non-Jews are largely prohibited, except for when they can sidestep those regulation -- but noting Rabbi Meir's harsher views in general regarding non-Jews, perhaps because of his lived experience. Note the distinction between monetary and personal treatments. Plus, the concern of Jew going to a non-Jewish barber in a private domain. Likewise, what happens when a Jew has business or medical dealings with a heretic - which is harshly cautioned against?