There is a sefer called קושיות ספר, written by one of the holy Rishonim (a student of the Maharam of Rottenberg). He writes a wondrous idea (ה"קכ אות). "You might ask, why do the goyim call the Jewish people dogs?
The answer is that it states (Devarim 14:1) 'לה אתם בנים אלקיכם, "You are children to Hashem, your G-d." בנים is gematria כלבים (102), dogs. Also, it states (Vayikra 25:55) ישראל בני לי כי, "For Bnei Yisrael are My servants," and the first letters spell כלבי, My dog."
Chazal (Avos 3:14) say, בנים שנקראו ישראל חביבין למקום, "The Jewish people are beloved for they are called בנים, Hashem's children."
The Gemara (Shabbos 155b) states, דעניא לית מחזירא דעתיר ולית ,מכלבא, "There is nothing poorer than a dog and nothing wealthier than a pig." Rashi explains that a dog doesn't have much food, and it is often hungry. Therefore, it is the poorest animal. In contrast, a pig always finds food, so it is called wealthy. This is Rashi's explanation. But we still wonder, why does the Gemara tell us this? What lesson are we supposed to learn from this information?
Kli Yakar (in his sefer שמיני ,גבורים עיר) explains that מכלבא דעניא לית, "There is nothing poorer than a dog" refers to the Jewish nation. They are called כלבא, "dog" because it states (Vayikra 25:55) עֲב ָדִים יִשְׂרָאֵל בְנֵי לִי כִּי, "For Bnei Yisrael are servants to Me." The roshei teivos of נֵיבְ ִיל יכִּ spells כלב, dog. Chazal says that the Jewish nation is the poorest of all. This is because they don't always have food. Most foods aren't kosher. When they come to a non-Jewish house, they can't eat. Even if the food itself is kosher, since it was cooked in a non-kosher pot, the food cannot be eaten.
The Kli Yakar writes, "This is the praise of Bnei Yisrael..." and he elaborates on this point. Surprisingly, we have seen from the Kli Yakar and from קושיות ספר that the Jewish people are called כלב (and they prove it with pesukim). We wonder, is there no better way to describe Bnei Yisrael than as a dog?
But based on our discussion above, the answer is simple and beautiful. The Torah wants to tell us that even when a person feels that he is on a very low level, like a dog, he is still a Yid, and Hashem says about him אלקיכם 'לה אתם בנים, that he is Hashem's child. Hashem says הם עבדי, they are My servants. Hashem says, "You are mine, and I want you."
Moshe told Pharaoh that they have to go three days into the desert to serve Hashem. Moshe explained (Shemos 10:9) נֵלֵ ךְ וּבִזְק ֵנֵינוּ בִּנְעָר ֵינוּ ל ָנוּ 'ה חַג כִּי נֵלֵ ךְ וּבִבְקָר ֵנוּ בְּצֹאנֵנוּ וּ בִ בְ נוֹת ֵ נוּ בְּב ָנֵינוּ, "With our youngsters and with our elders shall we go; with our sons and with our daughters, with our flock and with our cattle shall we go, because it is a holiday of Hashem for us." At first, Pharaoh didn't agree to this entire list. There were times that he didn't want anyone to go out of Mitzrayim, and there were times that he didn't want the children to leave Mitzrayim. But after makas חושך, Pharaoh finally agreed to the entire list, except for the flock and cattle. As it states (Shemos 10:24) וַיִּקְרָא יֻצָּג וּבְק ַרְכֶם צֹאנְכֶם רַק יְהֹוָה אֶת ע ִ בְ דוּ לְ כוּ וַיֹּאמֶר ֶׁהמֹש אֶל פ ַרְעֹה עִמָּכֶם יֵלֵ ךְ טַפְּכֶם גַּם, "Pharaoh summoned Moshe and said, "Go – serve Hashem. Only your flock and cattle shall remain behind. Even your children may go with you." Pharaoh finally agreed to Moshe's request that all Yidden may leave Mitzrayim. He was just stubbornly refusing to allow the animals to leave. Therefore, Hashem brought on מכת בכורות. We wonder, why was it necessary to make this great makah? Pharaoh more or less agreed to Moshe's request. Why was it so important that the animals also leave Mitzrayim?
Reb Shlomo Kluger answers that this was to teach a lesson to the generation before Moshiach. There are people in our generation who have fallen to very low levels, r'l, and they have reached the level of animals. Hashem showed them that He is ready to perform miracles and wonders to save the animals of Bnei Yisrael from Mitzrayim. Therefore, the people at low levels today should know that no matter what level they are at, Hashem won't forget them. Hashem will redeem them.
It is now the period of Shovavim, and Hakadosh Baruch Hu calls out to every Yid שובבים בנים שובו, requesting from us to do teshuvah. It states בנים שובו because we are Hashem's children, and a father never forgets his child. Even if a person feels that he is on the level of a כלב (gematria בן) or on the level of a בהמה (also gematria בן) Hashem wants him to return to Him. Hashem tells him בנים שובו, that His children should return to Him.