I wouldn't even put his name on the address label.
He's just a boyfriend.
He can sign the inside of the card.
But knowing how cards go, it is always the woman that does the cards anyway and signs the names for everyone. Most of the time.
And, ordering address labels and having it printed, does not have to be done. She can simply hand write the return address on the back flap of the envelope as the others have suggested too. And use just first names. Last names are not necessary.
And, since she doesn't want the not so close friends and relatives to know she is living with this man, then just don't send cards to them with his name on it. And simply hand write the return address on the card envelope.
The Boyfriend is not family, and unless he is actually paying for the cards/envelopes/stamps for the Christmas cards and cares about it, I would think most men/Boyfriends don't really care... if their name is on the card envelope or in the card anyway.
And if this is a boyfriend of only like 3 months, I don't think its important to have his name on the return address.
But if this is a boyfriend of say 2 or more years, then it is more relevant.