Although not explicitly mentioned in the Shuhan Aruch, the Kabbalists mention that each one of the three Shabbat meals corresponds to one of the Avot. According to Rabbi Avraham Azoulay (Or Hahama, vol. II, pg. 104c), each meal corresponds to the time of the day of the prayer that each of the Avot instituted. Thus, Friday night is Arvit which corresponds to Yaakov, the Shabbat day meal is related to Shaharit, which was instituted by Avraham, and Seuda Shelishit comes after Minha, which was instituted by Yitzhak Avinu. Nevertheless, Rabbi Moshe Cordovero (quoted in Or Hahama) and most of the Kabbalists order the Avot differently and this is the accepted custom. The Friday night meal corresponds to Yitzhak Avinu, since the night is a time of Judgment and Yitzhak has the related attribute of Gevura. The daytime meal, as well, corresponds to Avraham. Seuda Shelishit Kabbalistically joins the energy of the night and day meals, just as the attribute of Tiferet of Yaakov joins the attributes of Avraham and Yitzhak. Many have the custom to say “Birshut ______ Avinu” prior to reciting the blessing of Hamotzi of the corresponding meal.
Interestingly, the Arizal (Pri Etz Haim, Sha’ar 18, ch. 17) writes that the Shabbat night meal saves us from the birth pangs of Mashiah, the day meal saves us from the agony of Gehinom, Seuda Shelishit saves us from suffering in the grave. Furthermore, he writes that Melave Malka, the meal eaten after Shabbat and which corresponds to David Hamelech, saves us from the war of Gog Umagog.
Another detail regarding the Shabbat meal involves dieting. Rabbi Ben Zion Abba Shaul (Or LeZion, vol. II, ch. 21, § 3) says that one should not refrain from delicious food on Shabbat as this can take away from Oneg Shabbat, the pleasure that one must experience on Shabbat. However, if a particular food is considered dangerous to one’s health or if one may come to overeat, then the Shulhan Aruch (O.H. 288:2) writes that refraining from such a food or from overeating is considered one’s Oneg Shabbat. Furthermore, a vegan or vegetarian can refrain from eating fish or meat on Shabbat, since even though these foods are a source of Oneg Shabbat, they are not an obligation. The Arizal (Pri Etz Haim), however, writes that there is Kabbalistic benefit to eating fish on Shabbat.
Summary: The first, second and third meals of Shabbat correspond to Yitzhak, Avraham and Ya’akov, respectively. If one is dieting one should still partake of delicious food, but may refrain from eating food that is deleterious to one’s health, or from overeating. One may refrain from fish or meat if one does not normally eat such foods.