I think it's possible to eat healthfully on food stamps. That said, I think many don't. I believe that one contributing factor is access. Low-income areas are often served only by mini-marts with limited and generally unhealthy selections of food. I was pleased to see that Aldi has two locations in the city itself.
Another factor: cooking good food takes time. Processed food is convenient and well-marketed. This, indeed, is a health factor in the US at all levels of society.