"Don't punk out on me" usually means that someone is going to renege or back out on what they agreed to do because they're scared. A scenario might be two friends going to a haunted house. One says something like, "come on. Let's go," and the second person might show some sign of being scared or like they're going to run away and the first person says, "don't punk out on me." Another phrase that's similar would be to "chicken out."