The meaning is as if "not" modified "run", but syntactically "not" is attached to "must", I think. You can probably convince yourself of that by considering a sentence such as: they can yet must not run. So you are absolutely right to be confused by such sentences. There is something weird going on.
"Mustn't" means exactly the same thing as "must not" (but makes it clear that the negative should be scoped to the verb, by using an inflected negative form of the verb itself), so yes, there's something funny going on with "must".
See wiml's sibling comment for a correct reason why someone might be confused by such sentences. "Not" negates, so it is definitely not modifying "must". Clearly the word being negated is "run".
