Should women eat in the Sukkah? Yes they should. Are they required to? No they are not. They should also sleep in the Sukkah, as should men.
Halacha 1 Women, slaves, and minors are freed from [fulfilling the
mitzvah of] sukkah. A tumtum and an androgynous are obligated because
of the doubt [concerning their status]. Similarly, a person who is
half slave and half free is obligated.
A minor who does not require his mother's [presence] - i.e., a child
of five or six - is obligated [to fulfill the mitzvah] of sukkah
according to Rabbinic decree, to train him in [the performance of]
Halacha 6 We should eat, drink, and sleep in the sukkah through the
entire seven days [of the festival], both during the day and at night.
It is forbidden to eat a meal outside the sukkah for the entire seven
[day period]. However, [there is no prohibition] if one eats a snack
the measure of a k'beitzah or less, or even slightly more.
One may not sleep outside the sukkah at all, even a brief nap. It is
permissible to drink water and eat fruit outside the sukkah. However,
a person who follows the stringency of not drinking even water outside
the sukkah is worthy of praise.
Source: Rambam Laws of Shofar, Sukkah, and Lulav