There is not only no such thing as "illegal immigration"... in fact, there's no such thing as "immigration" at all!
For "immigration" to be a real thing you'd be claiming that borders and the "tax" farms they surround have legitimacy. You'd be claiming there is something "above" private property to "immigrate" to.
All there is with regards to this is migration and trespassing. Each individual who is moving on the surface of the planet is either within their rights to be where they are, or they are trespassing on private property.
If property is privately owned, you either get permission to enter, or you are a trespasser if you enter it anyway.
Government- the State- can own no property or anything else, since it possesses nothing it did not either steal or "buy" with stolen money, and thieves don't own the stolen property they possess. Government has zero "authority" to control who you let on your property.
So, "immigration" is a non-issue. You either trespass or you don't.
I am against trespassing. I am also against government pretending it has authority over other people's property (which is theft). I might choose to allow people to enter my property. I might not. Where they were born doesn't figure into that at all, and certainly not whether they have State permission.
If private property rights prevent individuals from going where they want to be, that is just too bad. (That also applies if your private property is surrounded by private property whose owners refuse to let you cross to get things you need to survive. I see that as very unneighborly, but it's just the way it is.) If private property rights prevent government goons from stopping "immigration", that is also just too bad.
I'll need to remember that next time the topic of "immigration" comes up.