You see, the point of sovereignty is that you make and enforce your own rules. We've become very lax on immigration, and when even Arizona just wanted to *enforce* the law, we had a foreign nation filing suit in our courts against us, and citizens of foreign nations demonstrating and protesting in the streets. We even have armed foreign military units crossing into our lands without our permission. Think--would you think of, say, Canada, as a sovereign nation if parades of people, crime cartel members, and foreign troops just crossed their borders whenever they felt like it? When a nation doesn't control who crosses it's borders and gives that power away by not using it, it loses some of its sovereignty.
Also, what other real nation in the world allows millions of illegal immigrants to live in their country? Why should we have to allow illegal immigrants while nations like Mexico have made it a felony punishable by years in prison?
Imagine if you live in an apartment building with 50 units. Now imagine that the guy who owns the building declares that you can't lock your doors. You can't prohibit people from just walking in and using your shower and sleeping in your bed whenever they feel like it.
I feel that's a semi-appropriate comparison.