Undocumented immigrant is far from a neutral term. They haven't just forgotten their documents, or oops! the government hasn't gotten around to documenting them. They are in the country when they aren't supposed to.
If you really hate the term "illegal immigrant", fine then call them unauthorized immigrants, but the "no person is illegal" slogan is just a motte and balley.
Again, motte and bally.
You also committed a logical fallacy:
> No person is illegal as borders are arbitrary
That's fallacious. All laws are arbitrary, as the GP pointed out, and you somehow failed to read:
> Speed limits are also arbitrary, enforced by other humans
So, it's factually true that borders are both necessary due to nature, and exist in the legal realm such that "illegal immigrants" is a factually correct label.