What an interesting idea! I would argue that you have to be able to read well to make good food decisions. Food packaging is intentionally designed to mislead you. You have to be quite savvy to understand what is really in the food.
One thing you could do is make up a large nutrition label, like you see on packaged foods. Have the students record several stats off their daily meals, to the best of their abilities -- carbs, fats, proteins, fiber, sodium -- and at the end of a week tally them up and show the students on the large label how much of everything they had eaten -- x pounds of sugar, x pounds of salt, etc. The link then would be the ability to read labels (literacy) and to analyze content for healthy vs. unhealthy.
You could add a pedometer to this task -- I'm sure the school would spring for some cheap ones, or you could see if any stores or organizations would be willing to donate them. Calculate the step ratio to calories burned, and see how it correlates with the amount of food they eat weekly.
For #4, Supersize Me is a great comedic documentary, but it is very subjective. The experiment has been replicated in many places and didn't always return the same results. It would be a good tool to spark discussion, but I wouldn't rely on it for unbiased, objective fact. However, it might help students cut down on their levels of fast food, or at least start choosing menu items with a wiser eye.
I have a similar idea to the one that is espoused in #2. Could you do an experiment or a test where half of your class has to engage in some form of exercise either before a class with you or before the school day, and the other half must do nothing except veg on a chair or something. Could you then ask students to reflect on their day or class and how they think it went, trying to consider how the exercise they did or didn't do played a part? Or could you get all of your students to do exercise before school one day and then no exercise the next?
On a slightly different note, I had great fun doing a unit on non-fictional texts by showing my students the documentary of Super Size Me. It is a hilarious documentary recording one man's pledge to eat nothing but fast food for an entire month, and the havoc he wreaks on his body as a result. It doesn't specifically fit your criteria, but it is a really thought-provoking film nonetheless.
Nutrition is a science, as is exercise, and the study of it requires more than simple participation, effort and coaching. To really achieve and practice habits of nutrition and exercise requires one to be a student of those things, both in and out of classrooms. From reading labels to reading health and exercise periodicals to becoming a physical therapist or personal trainer all require literacy and the discipline of practicing literacy. I think the connections are more natural than one might think.
Short of feeding them grade level articles to read about the statistical analysis of how literacy is affected by healthy nutrition and exercise habits, my thought, especially at the 6th grade level, would be to conduct an experiment. First, it would be difficult to call it a "literacy" test, as time likely does not permit an accurate measurement of each child's literacy, but for your purposes, maybe you could look at basic reading comprehension.
The basic idea would be that they read a series of short grade level pieces of text (you would have to find several that were similar enough in length and level for this to work) and take a basic comprehension quiz on each when they are finished. Perhaps they do one first thing in the morning, every morning. Though not perfect, it could be considered the control. Have them document what foods they ate the night before each reading quiz, and track their results in a chart. You might have them measure both how long it takes to read the text, and how well they do on the quiz at the end. You could conduct a similar experiment later with exercise/activity level rather than food as the variable factor.
I can't predict what kind of results you will get out of this experiment, and likely there is some detail work that might need to be looked at, but I imagine you could do this throughout a 4 or 6 week unit and have enough data to conduct some pretty interesting discussions. If nothing else, the class could look for patterns between types of food consumed and high or low scores. Additionally, you are going to pull in several inter-disciplinary skills (reading/comprehension, math, science, health, etc.) into one activity, which is always a bonus.