Even if we assume that a nation cannot exist without borders (itself a contestable claim because many nations have historically had unclear or contested boundaries), it does not follow that the maintenance of borders requires immigration restrictions. In reality, borders have a wide range of other functions, besides regulating immigration. For example, they define the territory within which a given government’s laws are binding, and also the land area within which it may deploy its armed forces without getting permission from other governments. If all immigration restrictions were abolished tomorrow, borders could readily continue to facilitate these and other purposes. A nation that doesn’t exclude peaceful migrants can still bar invading armies.
If we take Trump’s theory (and others like it) seriously, the Declaration of Independence did not make the United States a nation because it did not establish any immigration restrictions. Even worse, it condemned George III for “obstructing the laws for naturalization of foreigners [and] refusing to pass others to encourage their migration hither.” Instead of celebrating Independence Day on July 4, we should commemorate the enactment of the Chinese Exclusion Act. Jefferson Davis and his friends need not have taken the trouble of trying to secede from the United States in 1861. They should instead have argued that it simply did not exist in the first place.
UPDATE: I should emphasize that this post addresses only the claim that the inherent nature of nations requires them to impose immigration restrictions. It does not consider claims that immigration might have negative effects on nations (without causing them to cease existing), or even exclude the possibility that there might be extreme situations where the negative effects of immigration are so great that the nation in question collapses. The validity of such claims may vary from case to case, and certainly is not inherent in the definition of what it means to be a nation. Moreover, even where immigration causes some negative effects, it is often possible to mitigate or eliminate them without restricting migration itself (which is not true of claims that immigration restrictions are inherently necessary for the existence of a nation).
UPDATE #2: It is worth noting that claiming that nations can’t exist without immigration restrictions is not the same thing as claiming they can’t exist without restricting freedom of movement in some way. There are a variety of situations where restrictions on the movements of foreigners might be justified in cases where we would also be justified in restricting the movement of comparably situated native-born residents. For example, governments are often justified in restricting the movements of terrorists, violent criminals, spies, and carriers of contagious diseases, regardless of their nationality. Immigration restrictions, by contrast, forbid entry simply on the basis of where the person in question was born or resides.