No. The land that the Mexicans with indigenous ancestry are indigenous to, is in modern day Mexico.....not the US. They don't have Native American ancestry.....they have Native Mexican ancestry. Native American is a legal term defined as the citizens of tribal Nations within the US. Not Canada. Not Mexico. Not Peru. etc.
I am a Native from Canada.......while I legally do have the right via the Jay Treaty to live and work in the US, should that treaty not have ever taken place, then I would have no rights to live in the US, even though my Nation spans both sides of the Canadian/American border. Things are different now that there are modern borders in place.
Even before Canada, Mexico and the US were formed, one Native Nation wouldn't just move into another's lands, without expecting resistance or a fight. So no, Mexicans......especially since many are not even aware of which tribe in Mexico their Native ancestry may have come from......do not have any right to live illegally in the US, and claim it is because they are indigenous.
Whether or not illegal immigration is a hypocritical issue or not, is another question entirely.
Ojibwe....from Canada....so not a Native American either.