It depends on what you mean by healthy.
Organic food is hard to find at Disney (I think a few signature places have organic options).
Its hard to find whole grains. A lot of sit down places will have the option, harder to get a chicken sandwich on a whole grain bun at counter service.
Most places will serve salads. Most will serve some sort of leaner chicken dish. Milk and bottled water (WDW is a Coke partner - so its Dasani) is always available - but again, it isn't organic milk.
Vegetarians will always find an option, although they may wish to check menus over to pick places there the option is one they find appealling.
Disney is not all burgers, fries and chicken fingers (you can always sub carrots or something for fries) - but neither is it a place where you are going to be able to load up on your orange and dark green leafy veggies at every meal. I know a lot of people are unhappy with the "new healthier" menus (particularly for kids) - and I know a lot of people who wouldn't put anything bought at Disney into their body (those types don't travel much at all - and when they do, they buy their own food and prepare it).
You can bring in food. People even pack in their lunches.