Yes, it has two meanings. In the first sentence, he eats "on the run", meaning he eats while moving or while hurrying.
In the second case, if you are "on the run", you are a fugitive. The key difference is that in the first sentence, there is an ACTION VERB(eat) before the phrase. This is how you can tell which meaning it has. If someone IS (be-verb) on the run, they are a fugitive. If they are doing something on the run (eat, sleep, brush your teeth), then they are in a hurry.