I think it could go either way and that a person could definitely eat healthy or unhealthy on the same small budget.
I'm a full-time grad student. I was incredibly broke the past 2 months. I ended up eating less and very healthy during that time. I effortlessly dropped about 3 pounds of the 8 I had regained during the prior two years. I call it my Po' Diet. I'm considering marketing it in a book.
I'll soon be working full-time again. With the extra funds, I'll probably start buying as much organic meat as I can afford, so that will bump up my expenses a lot. But besides that, eating healthy doesn't really cost me more.