In Hayom Yom (22 Adar I)Hayom Yom (22 Adar I) the Rebbe quotes the Rebbe Rasha”b (Meheirah Yishamah 5648) as saying, “The obligation to eat Shalosh Seudos is learned from the Passuk “Hayom Lo”, therefore we don’t need to eat bread, but one must taste something. As Rabbi Yossi said (Talmud, Shabbos 118b) “May my portion be among those who eat Shalosh Seudos”.
So Minhag Chabad is to make sure to taste something for Shalosh Seudos, and the reason why they don’t wash for bread is because of the Posuk “Hayom Lo”.
What this means is as follows: The Gemarah in Shabbos learns out the obligation to eat 3 meals of Shabbos from the following Posuk: “And Moshe said, eat [the Monn] today (Hayom), because today (Hayom) is a Shabbos for Hashem, today you will not (Hayom Lo) find [the Monn] in the field”. Since the Pasuk says the word “Hayom” 3 times (when speaking about the Monn which served them as bread), we learn that one must eat 3 meals on Shabbos.
Based on the words of Rabbeinu Bechayei the Tzemach Tzedek rules (Piskei Dinim Chiddushim on Rabbeinu Yerucham p. 357) that since this meal is learned from the words “Hayom Lo”, unlike the first two meals which are learned from the word “Hayom”, we don’t need to eat bread for Shalosh Seudos, unlike the first two meals.