The Halacha, of course, permits a Jew to cook for a critically ill person on Shabbat. Obviously, it is also permitted for a non-Jew to cook for him. Only the patient may partake of that food on Shabbat. The question, however, is whether the food prepared by the non-Jew may be eaten on Mosa'eh Shabbat, by the patient or someone else. Since, it is now permitted to cook new food by a Jew, does the fact that the food is Bishul Akum (cooked by a non-Jew) render it forbidden even to the patient, or does the fact that it was prepared with permission in order to save a life render it permitted even after Shabbat? The Mishna Berura, in Siman 318, cites the Vilna Gaon that it is permissible, although in Siman 328:63, the Mishna Berura seems to prohibit eating the food.
Hacham Ovadia points out this seeming contradiction and suggests that the lenient position was written by the Hafetz Haim himself, whereas the strict opinion was written by his son. In this case, Hacham Ovadia gives weight to the son's position and prohibits eating the food on Mosa'eh Shabbat.
However, this only applies to a case where the non-Jew cooked the food in his own house. If the food was prepared by the non-Jew in the Jew's house, Hacham Ovadia presents a Safek Sefeka (Double Uncertainty) to be lenient: First, perhaps the Halacha is in accordance with the Poskim who permit the food because it was cooked with permission to save a life; second, even if the Halacha would not permit Bishul Akum when there is an alternative, here, perhaps it is not considered Bishul Akum, as the Halacha may be in accordance with the Ra'avad who rules that there is no prohibition of Bishul Akum when cooked in the Jew's house.
Thus, there is an interesting case in which a healthy Jew may eat Bishul Akum-when it was cooked by a non-Jew in his house on Shabbat to save a life.
Hacham Ovadia also discusses whether the pots used by the non-Jew to cook for the patient become un-Kosher, because of the Bishul Akum absorbed in them. He concludes that there is no problem, based on a Safek Sefeka. First, perhaps the Halacha is in accordance with the opinion that the Hachamim never extended the prohibition of Bishul Akum to utensils; second, even if the prohibition does apply, perhaps the Halacha is in accordance with the Ra'avad's position that Bishul Akum is not relevant to food prepared in a Jew's house. Furthermore, maybe the Halacha is in accordance with the Vilna Gaon and Mishna Berura that there is no problem whatsoever with food prepared in the context of saving a life.
This is the conclusion of Hacham David in Osrot Yosef, Siman 4.
Food cooked by a non-Jew in his home for a critically ill patient on Shabbat, may not be consumed on Mosa'eh Shabbat-neither by a healthy person or by the patient, as it is now permitted to cook new food. However, if the food was cooked by the non-Jew in the Jew's house, it is permitted on Mosa'eh Shabbat.
The pots used by the non-Jew on Shabbat do not Koshering.