1-Has anyone ever seen him not upset?
2-Has anyone ever seen him not be upset?
3-We have seen him not upset once.
4-We have seen him not be upset once.
I think '1' and '3' might work in an informal context.
'2' and '4' do not sound right to me. In them, 'be' seems to me to mean 'get'; whereas it is supposed to refer to being in a certain state.
Sentences 1 and 2 are the same, and the 'be' means 'be,' as in 'in the state of being upset.' If you wanted to mean 'get upset,' you could just as easily ask "has anyone ever seen him not get upset?" (which, when I think about it, is probably a more usual or natural way of asking #2).
Sentences 3 and 4 are also equivalent to each other, meaning that he's been upset every time we've seen him except once. And the same thing applies here, if you mean 'get upset' as opposed to 'be upset,' you can just replace 'be' with 'get.'