Generally, health codes do not permit food in the serving area of a licensed food serving facility that wasn't served by inspected site. There are a couple of reasons, mostly to do with cross-contamination and identifying the source of a food-borne illness. ("I got sick eating in the United Club!" [eating an egg salad sandwich I made at home and left in the car all day])

Yes, there are plenty of arguments against this and cases where it would be okay. But rather than having to deal with a big list of exceptions and arguing with everyone over them, it's easier for them to just say "No outside food or beverages".