The Gemara in Masechet Pesahim establishes that if dough is kneaded with fruit juice, as opposed to water, it cannot become Hames; unlike water, fruit juice does not have the ability to cause dough to become leaven as defined by Halacha.
Most Rishonim (Medieval Talmudic scholars) understand this ruling to mean that dough kneaded with fruit juice cannot become Hames at all. Rashi, however, in his commentary, explains that the dough in this case cannot become Hames "Mi'de'orayta" according to Torah law. On the level of Rabbinic enactment, however, fruit juice is equivalent to water in its ability to cause dough to leaven.
The Shulhan Aruch, in Siman 462, accepts the majority position, which maintains that fruit juices do not cause dough to become leaven at all. Thus, he writes, one may eat Masa prepared in this manner which is commonly referred to as "egg Masa" on Pesah, provided that he is certain that not a single drop of water was mixed into the dough. If water is added to a mixture of dough and fruit juices, the dough becomes Hames immediately. Thus, partaking of egg Masa is permissible only if it was prepared under strict supervision, such that it can be ascertained that no water was mixed into the dough.
A number of later authorities follow the Shulhan Aruch's lenient ruling, including the Peri Hadash (commentary by Rav Hizkiya Da Silva, 1659-1698) and the Hida (Rav Haim Yosef David Azulai, 1724-1806). The Hida adds that his grandfather, Rav Avraham Azulai, testified to the fact that the Jewish communities in Spain had the custom to eat egg Masa on Pesah.
Ashkenazim generally follow the ruling of the Rama (Rabbi Moshe Isserles, Poland, 1525-1572) who held that one should satisfy the stringent view of Rashi and abstain from egg Masa on Pesah. The practice of the Ashkenazim is to allow egg Masa only for the elderly or ill patients who find it difficult to partake of standard Masa.
It should be noted that Pesah is customarily a time when people follow stringencies that are not required on the strict level of Halacha. Accordingly, even a Sepharadi who wishes to follow the stringent view and refrain from eating egg Masa on Pesah may certainly do so, and regarding such a person it is said "Tavo Alav Beracha" ("he is deserving of blessing"). According to the strict Halacha, however, Sephardic practice, as found in the work Hazon Ovadia written by Hacham Ovadia Yosef on page 118, allows eating egg Masa during Pesah, provided that it was prepared under strict supervision.
Summary: Sephardic custom allows eating egg Masa on Pesah, on the condition that it has reliable Kashrut certification. Nevertheless, those who wish to be stringent, in accordance with the general custom to be stringent on Pesah, may certainly do so. Ashkenazim permit eating egg Masa on Pesah only for the elderly and ill patients.
