Actually, maybe I can be the voice of dissent here. I know it is a new day over there by now, but it sounds to me that your problem isn't that you are embarrassed or afraid to eat alone. Sounds to me like the real issue is that you feel lonely, hate the fact that you are stuck on a weekend a place you dont want to be doing something you dont want to do. All the rest is just ways of covering up that fact that you don't want to admit to yourself. As much as we like to boast that none of the rest of us have been there, the fact is I think most of us have.
So, for next time, or for when anyone else is feeling like this, here is what I do.
1) figure out what kind of comfort food you most want - forgetting completely about the national cuisine, and go get takeout.
2) also find a place that sells really good sweet things for desert that you can pick up too
3) a bootle of wine, beer, or sugary drink - whatever comforts you most
4) IF you can find a drug store like a CVS or something, most cities have something, pick up a few of those battery operated candles.
Then go back to your room, set up the candles, play some relaxing music on the computer, maybe take a bath, and just chill to yourself - take a vacation from world for a few hours. Face the clients - tomorrow. Just focus on yourself unwinding tonight.