First of all, you have to decide what healthy food is. There are many differences of opinion there and you have to separate the fad from the fact. How many people here kept hearing “eat your liver, it’s healthy” for you while growing up? By taking a middle ground, emphasizing fresh ingredients rather than processed (or recognizing what sort of processing is better or worse; I just saw a commercial lumping frozen vegetables with fast food as “bad food”).
Second, you run into the time sense problem. A lot of unhealthy food, particularly heavily marketed unhealthy food, contains strong flavors. These tend to overload the brain, which adjusts so that the intense flavors become “normal” and normal flavors become “underflavored”. While adults who realize this can eat what they consider to be underflavored food until they’re acclimated, children have far less patience, and will often reject the promise of future reward for current one. Therefore, you should expect a stage, probably lasting a month or two, where what you consider to be unhealthy food is forbidden. This will seem like a punishment to the kids, but, once they get acclimated to the more healthy food, they will find the unhealthy food doesn’t taste anywhere nearly as good as they remember. Keep the unhealthy food they still find edible for an occasional treat.