Is it true that the Chabad custom is specifically not to wash hands for (and eat) bread for shalosh seudos, and, if so, why?
In 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.
The reason is (perhaps) due to the fight between the two potential successors of the "previous Rebbe" of Chabad. Although there are older sources on the topic relating to the Rebbes, the widespread custom in Chabad was for the chasidim themselves to wash (according to the halacha), up until the mid-20th century.
"Finally since Gourary (the previous Rebbe's elder son-in-law) was delivering a chassidic discourse at the Shlosh Seudos meal in 770, the Rebbe abolished this ritual in Lubavitch. Of course Gourary continued to lead the 770 meal, but it was sparsely attended as all knew that this was without the Rebbe’s blessings." - R. Zalman Alpert (YU Reference Librarian)