SofiaAmes - I wrote a long post yesterday about the impact that my mother's rules about food and mealtimes have had, even to this day, on my relationship with food - please read it. I was more honest in that post than I think I have ever been about food and me - and I would hate for anyone else to end up feeling the way I do.
Also, as others have said, healthy eating is not neccessarily about three meals a day and no snacks. Children's stomachs are smaller so they may not be able to get enough food to provide their caloric and nutritional requirements if fed only three times a day - which is why people recommend more, smaller meals. This is also good for adults, as six smaller meals should keep the blood sugar at a stable level, avoiding the energy dips that leave us craving carbs/sweet things.
You seem to believe that eating more often during the day/eating snacks MUST mean eating crap/eating unhealthily/eating too much - and that is absolute rubbish. You spread the food/calories out between six meals instead of between three - it's that simple.
There is NOTHING harmful about a child (or indeed an adult) having a snack between meals, as long as their intake is balanced across the whole day. A slice of toast at 10.30am can be balanced out by a smaller portion of potato at dinner, for example.
And it is, I believe, very harmful to teach children that some foods are bad - creating forbidden fruit, as I said in my earlier post. Surely it is far better to teach them that there are some foods that are best eaten in moderation. An occasional slice of cake is fine - a slice of cake every day probably isn't.
My experience teaches me that it is essential to give your children a healthy relationship with food - and my fear is that the OP's approach is doing the opposite.