Native American DNA may stay in this country. Simple, and relatively cost effective. Those Native American descendants moving north will be fine. Yes, there has been some intermarriage with those of mostly Spanish ancestry, but so long as the DNA shows that they are also descended from indigenous peoples they would be welcome.
I find it comical that there is such hysteria about "illegal" immigration. It is the people who self identify as "white" who are most hysterical about "illegal" immigration. This landmass was populated by native peoples for many thousands of years before it was "discovered" by the English and Spanish. Then came the hoards of others of mostly Eastern European descent. All these immigrants were, "illegal." We did not invite you, but we were not hostile to you until you started killing us, stealing our lands, and making and breaking treaties with us.
Let's be clear about a few things. You stole Texas, Arizona, New Mexico, and California from the Mexicans. You claimed it was your "Manifest Destiny" to move from the East Coast to the West Coast, gobbling our land and "giving" us the least habitable lands on this continent, the poorest of soil, and without much water. You "gave" us the lands where little would grow and our livestock could not graze. You tried to exterminate us. Well we are now sick of you and we want you to go home. The only immigrants among you we will keep are those you captured, transported here in ships, and sold into slavery. These people are our brothers and sisters. The rest of you should start packing. Your time has run out. If you cannot claim ancestry on this continent (which does, by the way, stretch from Canada to Argentina) prior to the early 1700's consider yourself served with an eviction notice.