I always dress well when I fly Business Class (I never fly First).
Why? Because if am flying Business Class, I am flying on a friends-and-family coupon from someone who works for the airline, and the airline (for some reason) insists on a dress code for deadheads like me. (Not that kind of Deadhead, the not-high kind.) Slacks, not jeans; button shirt; socks. Last time I did this, I was flying to a funeral and the standard for the funeral was considerably lower.
If you are paying your own way, you can fly in your boxer-shorts, if you are so inclined.
Here is a discussion of United's dress code for us non-revs and it cites United's policy is full (although, without a link).