We have problems with ILLEGAL immigrants.

What, simply because it's illegal? Would you have a problem with the Underground Railroad if you were alive in 1850? If illegality is the problem, let's repeal the laws restricting immigration, end of problem (except perhaps for some transition effects).

But I suspect that you really mean you have problems with indiscriminate immigration; that you want some (potential) immigrants to remain illegal, you want the government to say: "You get in, you don't." Is the government really a better judge of who's desirable and not than, say, the people who would decide whether or not to have Felipe and Consuelo as employees or tenants? Do you really have zero problems with any of the government's past choices in this respect?

Do I have a right, morally, to deny you access to my house, even if you were born in the house next door?

You have a right to exclude me or Brad (or your house's previous occupants!) from your house, but no right to exclude us from Romana's house: only Romana has that right.

We have property rights in this country and it extends all the way to the border.

Then I choose to exercise my property rights by inviting any and all persons of goodwill onto that portion of my property that I am not personally using.

But jurisdiction is not the same thing as property. (When territory is transferred from one government to another the landowners therein are not automatically dispossessed.)