What is the origin of this custom? This must be fairly recent and "Lubavitch-specific", because my grandparents, who were very observant, always had matzah ball soup and fried matzah with eggs for Pesach.
R' Eizik Vitebsker writes (look in Os 26) that the origin of this Chumra was from the Mezritcher Maggid.
R' Shneur Zalman of Liadi explains that since some opinions say that flour which was baked (without being kneaded first) can still become chometz after contacting water as it may not have been baked well.
He writes that (at least in his time) one could see flour on his Matza after baking. Therefore, one should not place such Matza in soup as the flour will become Chometz.
He writes that even though there is technically no concern as the halacha follows the Rambam and Rashi who permit one to cook flour which was baked, one should still be stringent following the words of the Arizal "to follow all stringencies on Pesach".
This custom is known as gebrochts (Yiddish for "broken"); or "matza shruya" (soaked matza) in modern Hebrew. It's prevalent in many Hassidic and Hassidically-influenced communities, though many first encounter it with Lubavitch.
The custom arose out of concern that there may be a packet of dry flour in your matza. If that flour never reacted with water, then when you break your matza into your soup or the like, the flour+water could become chametz in a few minutes. (Whereas if you mix flour thoroughly with water and bake it fast enough into matza, it can no longer become chametz.)
The custom has been around several hundred years; it appears, for instance, in the Shulchan Aruch HaRav, a code of Jewish law by the first Lubavitcher Rebbe (author of Tanya) about 200 years ago.
Many families -- certainly Sefardic Jews, and many non-Hassidic Ashkenazic ones (especially from places closer to Germany or Lithuania, not Romania or Hungary) never adopted such a custom, such as your family -- and mine.
The roots of this minhag actually lie in the Gemara itself. In Pesachim 40b, there is a discussion which says explicitly that Rav Papi allowed servants in the beit Reish Galuta to thicken a tavshil with "chasisi." The Rif says this is matzah meal; Tosafot say it is lentil flour, and Rashi says it is dried flour. Rava says we need to be concerned in a place where the servants are not makpid on the mitzvah and may come to mix in actual flour, instead of the matzah meal. Rava himself did actually stir matzah meal into a tavshil in his own house, and was merely concerned when dealing with servants. The claim, however, that cooked matzah can become chametz gamur lacks halakhic grounding. The Tur, to Orah Hayyim 43, paskens that it is forbidden to cook with matzah meal when servants are around, but this itself doesn't account for the shita to avoid all kinds of matzah sheruya- wet matzah. The Chatam Sofer, Vilna Gaon, and others ate matzoh balls, and the Shulchan Aruch paskens that wet matzah can be eaten by cholim, and even rules that we can wet and rebake matzah. It seems to be based on a teshuva of the Alter Rebbe of Lubavitch, who notes that Rabbeinu Yerucham held that dried/roasted flour (as described as Rashi in Pesachim 40b) can become chametz when mixed with water (not normative halakha at all). Citing the Arizal's position that extra humrot are proper on Pesach, he was makpid on the minority shita and chose to avoid all matzah sheruya. He also notes that in his time, matzohs often had uncooked flour on the surface because people had recently become makpid on 18 minutes.
The reason for this custom is the suspicion that some flour remains uncooked in the matzah, and by water then touching it, allows for the opportunity of becoming chametz.
Matzah with water, or certain other liquids, is called "gebrochts" in Yiddish, and people who follow the custom to refrain from eating it are often said to "observe gebrochts" or "keep gebrochts".
The custom is prevalent mostly in those from Chassidic backgrounds, especially Lubavitch, since it is mentioned by the Baal HaTanya.
It's interesting that another answer mentions the Alter Rebbe's answer as the origin, but fails to clearly explain the history of how/why this minhag suddenly started.
Why isn't this minhag / worry mentioned by any of the poskim ??
Here is an excerpt of the Alter Rebbe's answer:
ומה שלא הזכירו זה בפוסקים, היינו משום שזה אינו מצוי כלל אלא בעיסה קשה שלא נילושה יפה, ובדורות הראשונים היו שוהין הרבה בלישה וגלגול עד שהיה נילוש יפה . עד שמקרוב זה עשרים שנה או יותר נתפשטה זהירות זו בישראל קדושים, למהר מאד מאד בלישה, ואין לשין יפה יפה, ולכן נמצא קמח מעט במצות של עיסה קשה, כנראה בחוש למדקדקים באמת .
Throughout Jewish history matzah was thoroughly kneaded. This process took time, and this was not a problem since the halacha states that while kneading the dough - it cannot become chametz. (see Rambam Chametz Umatza 5:13). Because of this fact, there was never any worry that after baking you'll see some flour on the surface of your matza, because all of the flour became dough.
Only in the times of the Alter Rebbe (20 years or so before the writing of his tshuvah) did the Jewish world decide to take upon themselves a chumrah: "Let's bake the Matza's from start to finish in 18 minutes". This chumrah is what led to the Gebrochs problem. Now that everyone hurried so much, they spent less time kneading the dough, and as a result the finished matza had a very small amount of flour on it.