Is one required to pursue a "Shomer Shabbos" or "Shabbos friendly" residency?
To the best of my knowledge, there is no posek who allows one to pursue a residency in which one will have to transgress even a rabbinical Shabbos prohibition. The prohibition of Sabbath desecration to become a doctor is accepted by the spectrum of Orthodox rabbinic authorities.
For example, see the ruling (along the right sidebar) of Rav Hershel Schachter, Rosh Kollel in RIETS' Marcos and Adina Katz Kollel (Institute for Advanced Research in Rabbinics) and occupant of the Yeshiva University Nathan and Vivian Fink Distinguished Professorial Chair in Talmud. Rav Schachter writes:
"And even after having completed his school years, the future doctor must take special care to make sure he has a Sabbath-observant residency. If this can not be arranged, the student must simply look for a different profession."
The most common source for the misconception that one may transgress Shabbos as part of a residency is from a ruling (see side or click on it to get it to download) of Rav Moshe Feinstein as recorded by Rabbi Moshe Dovid Tendler and Dr. Fred Rosner in Practical Medical Halacha, published by the Association of Orthodox Jewish Scientists.
The ruling states that "a physician must seek association with the most reputable and prestigious hospital possible to ensure excellent training and continuing education." But the ruling makes clear that "if superior training is to be acquired at the price of Sabbath desecration, even of rabbinic ordinances only, the student-physician must forego the educational advances of the prestigious hospital." There is no halachic sanction for being mechallel Shabbos even once to obtain medical training. Only if one is sufficiently familiar with ALL of the intricacies of hilchos Shabbos, and only as the above-mentioned responsa states, "provided that he is certain of his fortitude in maintaining all halachic requirements, despite the less favorable environment," may one even consider working in the hospital on Shabbos.
The responsum further requires that "Open discussion with the training hospital administration must be initiated before accepting such an appointment." This is a clear requirement to investigate any Shabbos conflicts before the match and to discuss issues of Shabbos observance with the program director before ranking any program.
That does not mean one must go to an official "shomer Shabbos" residency, but it does mean that one must be scrupulously shomer Shabbos, even during residency. In any situation in which melacha (forbidden Shabbos labors) is permitted, one is not desecrating Shabbos. But it is also important to realize that pursuing a medical career is not per se a permission to transgress Shabbos. I personally confirmed the above interpretation of the responsa (that one must sacrifice convenience of religious observance, but not the strict adherence to halacha to obtain better training) with Dr. Rosner in 1993.
It is because of this requirement of Shabbos observance, even during residency (and post-training medical practice), that the need for "shomer Shabbos" and "Shabbos friendly" residencies arises.