Even though I'm vegetarian and even though I would love to think that people could beat cancer by simply eating vegetables, that isn't the case so I don't know why we would come up with a phrase for it.
We can already say that eating a lot of red meat raises one's chances of getting cancer and that having a more healthy, plant-based diet once you have cancer can aid your recovery.
I see no need to come up with rather simplistic trite phrases for something so complicated and serious. People "fight their way through cancer" (or they don't and they die).
Interested in Language