The second phrase in the original question is not correct, it should be "eat your way to health."
So, the correct comparison is between "eat yourself healthy" and "eat your way to health." In my opinion, there is no substantial difference, nuanced or otherwise, between the objective meaning of these two phrases. In my opinion, however, "eat yourself healthy" is very awkward, and is likely to confuse most people who read it. It just sounds weird. It conjures up strange images of you "eating yourself."
"Eat your way to health," on the other hand, sounds perfectly fine, and the meaning (as I understand it) is perfectly clear. So if you are trying to choose between these two phrases, I recommend this one.
In summary, the nuanced difference between the two phrases is that "eat yourself healthy" sounds strange and vaguely obscene, while "eat your way to health" sounds nice and perfectly understandable.
UPDATE: Asker has requested comparison of "laugh yourself healthy" and "laugh your way healthy". In this case, "laugh your way healthy" is still incorrect, and should be "laugh your way to health."
What is different is that "laugh yourself healthy" does not sound particularly awkward or inappropriate. So, building on Max Williams' comment, the main difference between the two phrases is that "laugh your way to health" implies a journey to health as a destination, while "laugh yourself healthy" connotes a somewhat more direct effect of laughing in bringing health.
Althernatively, we could compare "laugh yourself to health" and "laugh your way to health". In this case, "laugh yourself to health" has a connotation of self-help, of you laughing and making yourself more healthy, whereas "laugh your way to health" is more indirect, and again feels a bit more like a journey that you take, through laughing, and eventually get to the destination of "health".
In any case, the objective meaning of all 3 correct phrases, "laugh yourself healthy," "laugh yourself to health," and "laugh your way to health" are the same, in my opinion. Hope this is helpful.