For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
Communion:
In the early Church, only those who received the Word and were baptised joined in the breaking of the bread (Acts 2:42,46). We read only of the "disciples" coming together to break bread (20:7).
The bread is the communion of the Body of Christ. Those who are not organs of the Body of Christ cannot partake of that bread (1 Cor 10:16,17). Only those who are born again or those who are baptised by the Spirit into the Body of Christ have a right to the Lord's Table. If those who have not become the Children of God by repentance and faith dare to participate in the Lord's Supper, those who are born again should not become part of it. We should not provoke the Lord to anger.
The Passover is a type of the Lord's Table. The Lord said to Moses and Aaron, "This is the ordinance of the Passover: No outsider shall eat it. But every man's servant who is bought for money, when you have circumcised him, then he may eat it. A sojourner and a hired servant shall not eat it" (Ex 12:43-45). Today the circumcision is of the heart. Only those who have experienced the miracle of new creation can enjoy the new covenant meal.
If in your Church or assembly the bread and wine are served to those who are not born again, have a chat with the pastor and request him to stop it. If he refuses, you have no other go than to partake of the Lord's Supper in another assembly where only believers are served. There are levels in fellowship and therefore such an arrangement to agree with one's conviction should not be difficult.
Can believers break bread with unbelievers?
Never! "What fellowship has righteousness with lawlessness? What part has a believer with an unbeliever?" (2 Cor 6:14-16).