The fact that it was BIL's birthday is irrelevant and a red herring. He and SIL were rude and greedy to take food that everyone had paid equally for, especially without even offering other people the chance to take their share.
I feel that if people wanted "their share" they should assert themselves and take it then. I don't think they need to be offered their share.
In my mind, this is one of those "it depends" situations.
If the process was something like the wait staff taking away the leftovers, boxing them up, putting the boxes in bags and then giving the boxes to your BIL - all this on their own initiative, not at BIL's request - then I can understand him just taking the bags. Once the situation is at that point, in the absence of someone speaking up and saying, "Oh, I had wanted to take some of that leftover fettucine home", it's the path of least resistance. And as TurtleDove said, it's on the people who want leftovers to make that clear.
If, however, the process was more like BIL asking the wait staff to box everything up and give it to him, well, yes, there is some rudeness there. At the time of his request, someone else could still speak up and say he/she wants to take something home, but it's awkward (at best) to basically overrule the guest of honour. Not rude, but awkward.
And if the process was that the wait staff boxed everything up, left the (unbagged) containers on the table, and BIL just took them all without saying anything, that seems rather rude to me as well. Although, again, at the point the containers were left, everyone else had a chance to speak up and claim something.