I'm going to agree with lbftw on that one, acutally.
This is coming from a middle school kid who's not a bully or anything, but typically kids don't sit alone in the cafeteria. So when other students see a kid sitting alone, they're going to wonder why. Either the kid is weird, no one wants to sit with him...they might pity them or whatever, even though OFTEN it's not true. Even as a little kid if you saw another person on the swings alone, you'd feel bad and go play with them, even if the person was just fine playing alone. So yes, h-o-p-e-f-u-l, it's ok to do things alone but other kids may not see that the same way, especially at meal time. With things like studying, watching tv, just grabbing a snack, etc, kids will be very understanding about. But, I don't know, something about the cafeteria just makes it different.
I don't mean to be rude, or exclusive, but as a student that's just my observation. I do encourage hopeful's second piece of advice though.