There is nothing wrong whatsoever with refraining from milk and/or meat for health reasons. If one does not want to eat these foods, it is possible to do without them on festivals as well. The Rambam (Maimonides) gave much advice regarding the healthy foods to eat. It is not for us to give advice about what is or is not healthful, but if you are convinced that a given kosher diet is wholesome (and have a reasonable basis for your conviction), then halacha looks on your efforts to keep healthy most favorably.
Let us now deal with Shabbat and Yom Tov.[1] Even though we have not merited in our time to have a Beit HaMikdash[2] and do not make offerings and partake from the festival sacrifices, we still have a mitzva to indulge in festive meals. On Shabbat, this is referred to as oneg (delight), on Yom Tov, as simcha (joy).[3] Classically, this includes eating meat and drinking wine (for children, it includes eating candies).[4]
However, if someone has a dislike for meat, he need not force himself to have it. If one prefers fish or some other food, it is halachically valid to use it as a substitute. What is crucial is to treat the meal with the proper festive mood, which depends to a great extent on the menu. The specifics depend on the particular person.[5]
If one enjoys meat but refrains from eating it for health reasons, then the matter depends on how important the dietary considerations are. If one anyway bends his dietary ‘rules’ for special occasions from time to time without significant consequences, then Shabbat and Yom Tov would also be a time to do so, at least in moderation. If one is strict about his diet, one may refrain from meat and find an appropriate substitute.