I am in charge of what goes into the food vending machines in my school, so I actually have to grapple with this question on an ongoing basis. It's so easy so just go with the knee jerk response of "yes, of course!", but the issue is not all that simple.
Are you including foods students bring from home? That's a huge question, and gets into personal rights and responsibilities. It also gets into issues other than the food itself. For example, a school near me made a rule change this year for all beverages; the only thing students are allowed to drink in school during the day is water, and it must be in a clear bottle. That sounds sensible enough, until you get into the plastics issue. I personally won't drink from plastic, even if it's BPA-free. I have a couple of (expensive) stainless water bottles, and that's what I use; if this rule went into effect at my school I'd have a problem with it.
Additionally, dividing foods into the categories of healthy versus unhealthy is not all that simple. As an example, let's consider beverages. Currently, according to federal hot lunch guidelines for beverages, schools are only supposed to sell water, 100% juice, milk, or drinks that use non- or low-calorie sweeteners. However research indicates that juice consumption can increase diabetes risk, and high juice consumption has some other health risks for small children. Artificial sweeteners have been positively associated with migraines, they have been shown to cause detrimental changes in amino acid metabolism, research is still underway on whether these substances promote cancer, and they may even be unsafe for diabetics. As you can see, I've done a lot of reading on them, and I personally believe that artificial sweeteners are extremely unhealthy, and I would never allow my children to have them. Who's to say what healthy or unhealthy foods really are?
I'd have to agree with #3, everyone would be much better served if we put our energies toward educating children on making their own healthy choices in all areas of life. Parents also need to be educated on how to afford and prepare healthier meals at home.