This is a difficult one for me. I'm glad that you asked this question because it has forced me to answer it for my self, as well as for you.
My husband is an omnivore. I am technically a vegetarian, even though I don't eat dairy, eggs or honey. The rest of the family is omnivores, although my Mother-in-law is a "Flexitarian." Additionally, most of the surrounding culture is omnivorous.
Question one: will you raise your children to be vegan?
I will encourage my children to eat all the good foods that I do, and explain to them why I don't eat meat. But I'm not going to kid my self into thinking that they will never have meat pass their lips.
Question two: will you give them a choice?
It is not so much that I will give them a choice as I know that they have a choice. Even if I don't "give" them a choice while they are living with me their will be the inevitable time when they are outside my sphere of influence and they are offered, or they seek, to do something contrary to what I have brought them up. This includes both my dietary and moral values, heck, even my fashion sense.
Question three: We all know how fickle children are what if they don't want to be vegetarian?
It's important to note that it takes up to 15 times of offering a child a new food for them to accept it, and about a hundred times of eating it for them to develop a taste for and like it. And if the parent and the rest of the family don't eat a food, it is unlikely that a child will ever eat that food, even if offered.
I remember quite vividly watching such programs as Sesame Street and Popeye the Sailor Man as a child. I can honestly say that those programs had an effect on my dietary choices and my love of vegetables. Spinach is my favorite vegetable. Just think, if a t.v. program that is around a child for only thirty minutes per day, five days a week, can have such a life long effect on a child, just think about what a positive effect a child's parents can have on them.
Question four: Do you think that that sort of diet is healthy for a child honestly?
I do think that if the very essence of a diet is healthy for an adult then, yes it is also healthy for a child. It is important to realize that children are not miniature adults. Their nutritional requirements are far greater than an adult's because they are still growing. Their requirements for food may vary significantly from day to day, month to month, year to year and even by the hour. At the same time they are absorbing the culture around them and learning social norms. That includes attitudes toward food. It is important not to force a child to eat too much, and no one wants to be responsible of feeding a child too little.
Regular meals and snacks, variety and balance in those meals, education on nutrition, manners and dental hygiene, all dispensed for the level of the child are all important. I also think that getting a child involved in cooking at an early age is important.
I hope that I have answered your question satisfactorily and thoroughly.