Pork and its products, alcohol, blood, and animals not slaughtered properly are forbidden, but Allah explicitly permitted the food of the People of the Book (Jews and Christians), so their meat is generally lawful. You may eat with and accept food from your non-Muslim family as long as the dish itself contains nothing haram like pork or alcohol; vegetables, seafood, and dairy are all fine. If you are unsure whether Allah's name was mentioned over a piece of meat, the guidance of the Sunnah is simply to mention Allah's name yourself and eat, without excessive suspicion. So there is no need to isolate yourself from your family's table over doubts.
Evidence — the daleel
On deen2u
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.