I know it sucks, and my husband would be furious with me too!
That being said--put yourself in the kids' shoes. He's at a stranger's house, his dad bailed on him and he can't be with his mom after Christmas. I'm sure he has issues.
Have a talk with the mom about the son's attitude. Let her know that if he doesn't turn it around, you won't be able to watch him. Believe me, Mom will have a talk with him if she's desperate.
Then tell him the rules of your house. Be firm but kind. When he oversteps by being rude, you can say "I'm sorry you don't like our toys, but that's what we do in our house and I'd appreciate it if you'd be a good guest." I have learned that kids need to be taught how to be good guests. My stepdaughter did not have a lot of playdates when she was young because of being bounced back and forth between houses. When she turned 7 I arranged playdates and was horrified at her manners! These things have to be TAUGHT, and often times single moms and dads are more concerned with other things than teaching manners. Because my stepdaughter was back and forth between mom and dad, when she was at one house she was the center of attention (because she wasn't there all the time) and the same goes with the other. She was spoiled because mom and dad felt guilty, and also they didn't get to see her all the time. This led to her manners EXACTLY like you described this little boy. She complained at other people's houses, once she was offered Kool Aid and she said "no, I want juice." Well I'm thinking the mom want to save the juice for her own kids and when she said "well, we're serving Kool Aid" my stepdaughter said "well you have juice, I saw it in your fridge and that's what I want." Oh the horror!
Try to enlist your husband in at least feeling sorry for the mom. Then be firm with the child and bring him along. It's okay to even pull the child aside and be more firm. I would even say "your attitude is ruining things for my family. Please change your attitude or I will have to phone your mom at work and have her pick you up."
If you hit a wall and find the attitude is the mom (who doesn't think there is anything wrong with HER child) then now you can bow out and not feel bad.
Good luck!