Since he was born in the U.S.

No and you're an idiot.

He should be eligible. Though I think its also possible if you become a legal citizen too.

He could when he's old enough. He is a U.S. citizen. He was born here and, not that it matters, his father is American as well. He'd need to renounce his British citizenship (he has dual citizenship) - I would think he'd have to renounce it anyway.

The root of all unhappiness is unmet expectation.

You have to be born here.