A basic principle of whether a given dough can be used for mitzva matzo is that dough that does not have the potential of becoming chametz by simply sitting for 18 minutes cannot be made into mitzva matzo. Thus, a dough made from juice or something similar is of doubtful validity as mitzva matzo and may be used for the mitzva only in cases of illness or age.
Those who contend that Ashkenazi Jews should not eat egg matzah cite Rema (Orach Chaim ibid., 4) ruling that the custom among the Ashkenazim is to refrain from using Egg Matzah on Passover at all, unless it is necessary for children or the elderly who would have difficulty eating regular Matzah.
Commenting on Rabbi Yosef Karo's permission to use egg matzah, the Rema responded "in our communities, we do not knead (matzah) dough with fruit juice. And one should not change from this unless in a time of emergency for the sake of a sick or old person who needs this". Those who follow this prohibition of eating egg matzah on Passover also include chocolate covered matzah, grape flavored matzah, and the many other varieties available.