What is the ruling on Muslim refugees eating beef and chicken not slaughtered in the Islamic way if this is the only food available to them from the governmental authorities that prepare it, and which cook pork and unslaughtered meats, and use the same utensils and oils to cook vegetables? And is this situation considered a necessity that permits them to do so?

The default rule is that it is impermissible to eat carrion or impure food. People must strive to find lawful (halal) and pure food. However, if they find only impure food and it cannot be purified, they are permitted to eat it out of necessity, based on the Almighty's saying: {He has only forbidden to you carrion, and blood, and the flesh of swine, and that over which any other name than that of Allah has been invoked. But if one is compelled by necessity—neither desiring it nor transgressing—there is no sin upon him. Indeed, Allah is Forgiving and Merciful.} They may eat until satiation if the necessity persists.