Oznayim LaTorah to Beshalach 16:25 says that the verse אכלוהו היום כי שבת היום לה' וכו - eat it today, for today is Shabbos for Hashem - means to say that they should eat the Manna on Shabbos, and not be like those who refrain from eating it (since the leftovers of previous days were spoiled) even though they see nothing wrong with it as it is. Therefore Moshe told them to eat it, as שומר מצוה לא ידע דבר רע, one keeping a Mitzvah will not know evil [from the keeping of the Mitzvah]. He adds, as a parenthetical statement, that this is a support for those who diet all week and then go off their diets for Shabbos.
See it here.
It doesn't necessarily mean it "won't make them fat" but it does mean that those watching their weight need not be careful on Shabbos. The reasoning is because in this eating they are involved in a Mitzvah. (Accordingly, it may only work if your eating is for the Mitzvah of enjoying Shabbos, and not just because you like to stuff yourself once a week.)