There is a question, intended to provoke thought, often asked of children - "Which came first, the chicken or the egg?". The point is that there is no simple answer. Either one leads to a dilemma of causation. Thus, "chicken-and-egg" is a metaphoric adjective to describe a situation where it is not clear which of two events is the cause and which is the effect. "American Gods" is a fantasy TV drama. Your character A is Mr Ibis, who is black (so-called "African American"), and is the ancient Egyptian god Thoth. He says that he does not think of himself and his fellow Egyptian gods as "African". It is then pointed out to him that he came from Cairo in Africa to Cairo in America. He replies that it is a "chicken and egg" situation. It's a joke. Egyptian gods are expected to say mysterious things, and the TV series, which has been called "dark comedy", is derived from a book by Neil Gaiman, whose works have been described as combining horror and humour.
In case you think I have not exactly explained the cause-and-effect nature of Thoth's remark, it may be worth considering the BBC reviewer's comment about the character:
hey – it doesn’t have to totally make sense.
But who is Ancient Egyptian god Thoth?