What is the significance between the struggles with Yaakov and Esav's physical body and the 4 exiles Daniel revealed from Nebuchadnezzer's dream?
... The first empire - the Babylonian Empire - sought the importance of being considered "everything", as it is written in Daniel (Daniel 2:38) "you are the head of gold";
But the second empire [Daniel] said about it (Daniel 2:32) "its breast and arms were of silver". For this empire loved the accumulation of wealth, and 'the lover of money will never be satiated by money';
And the third Empire [Daniel] saw it (Daniel 2, 39) "Copper", because it was bold and swift to go and conquer distant lands, and therefore it's empire was aggressive and bold like copper which is bold;
And the fourth, its power was in its destruction, and therefore it is metaphored (Daniel 2, 40) to iron, which cuts and destroys everything.
It seems Yaakov struggled with Esav in reverse of each of these body references:
At the birth of Yaakov, he grasped Esav's heel, which may correspond to the feet of iron and clay; the pig; the 4th and last exile of Rome.
וְאַֽחֲרֵי־כֵ֞ן יָצָ֣א אָחִ֗יו וְיָד֤וֹ אֹחֶ֙זֶת֙ בַּעֲקֵ֣ב עֵשָׂ֔ו וַיִּקְרָ֥א שְׁמ֖וֹ יַעֲקֹ֑ב וְיִצְחָ֛ק בֶּן־שִׁשִּׁ֥ים שָׁנָ֖ה בְּלֶ֥דֶת אֹתָֽם׃ Then his brother emerged, holding on to the heel of Esau; so they named him Jacob. Isaac was sixty years old when they were born. (Toldos 25:26)
The struggle between Yaakov and the arch-angel of Esav, which resulted in dislocation of Yaakov's hip (Gid hanasheh), which may represent the bronze thighs; the hyrax; the third exile of Greece.
וַיַּ֗רְא כִּ֣י לֹ֤א יָכֹל֙ ל֔וֹ וַיִּגַּ֖ע בְּכַף־יְרֵכ֑וֹ וַתֵּ֙קַע֙ כַּף־יֶ֣רֶךְ יַעֲקֹ֔ב בְּהֵֽאָבְק֖וֹ עִמּֽוֹ׃ When he saw that he had not prevailed against him, he wrenched Jacob’s hip at its socket, so that the socket of his hip was strained as he wrestled with him. (Vayishlach 32:26)
- The encounter between Yaakov and Esav, when according to the Ba’al Ha-Turim; Avot deRebbe Natan 30b, Esav tried to bite Yaakov's neck which turned to stone, which may correspond to the neck of silver; the hare; the second exile of Persia.
וַיָּ֨רָץ עֵשָׂ֤ו לִקְרָאתוֹ֙ וַֽיְחַבְּקֵ֔הוּ וַיִּפֹּ֥ל עַל־צַוָּארָ֖ו וַׄיִּׄשָּׁׄקֵ֑ׄהׄוּׄ וַיִּבְכּֽוּ׃ Esau ran to greet him. He embraced him and, falling on his neck, he kissed him; and they wept. (Vayishlach 33:4)
- The burial of Yaakov, when Chushim, son of Dan, beheaded Esav to allow his grandfather burial into the Ma'aras Hamachpela, which may represent the head of gold; the camel; the first exile of Babylonia.
... And Chushim was deaf and dumb, yet he could notice the voice of men in an uproar. And he inquired, saying: Why do you not bury the dead, and what is the cause of this great uproar? And they explained to him all concerning the words of Esau and his sons, and he ran towards Esau in the midst of the fight, and he smote Esau with a sword, and he cut off his head and it rolled away to a great distance, while Esau fell down dead amongst the fighting people. (Midrash Rabba Sefer HaYashar Vayechi 5)
Additionally, Rabbeinu Bahya equates these 4 kingdoms to the 4 unclean animals pointed out in the Torah (Rabbeinu Bahya, Vayikra 11:4):
Midrash Tanchuma Shemini 8 points out that four animals are mentioned here in four separate verses to allude to four different empires which will play host to the Jewish people in exile. Verse four speaks of גמל, the camel, a reference to the Babylonian exile.
Concerning the hare, ארנבת, mentioned in verse six, this is a reference to the exile under the Medes, (Ahasverus).
In verse five, the Torah speaks of the שפן, hyrax.” (it has short ears and hoofed feet). This is a reference to the exile under the Greeks.
The pig, חזיר, mentioned in verse 7 is a reference to the Roman empire, the fourth exile, the one which we still endure.
Just for reference, there is a picture circulating around the internet that depicts the statue Nebuchadnezzer saw: