All Native Americans, regardless of whether they live on tribal lands, are American citizens. When the question is asked whether Native Americans living on tribal reservations also have a separate citizenship, the answer is complicated, but mostly yes.
No, tribal lands are not their own “country,” (although some scholars use the language “domestic dependent nations” to describe their pseudo-sovereignty) but they are treated with a similar respect of sovereignty that other western democracies have established to (in a very small measure) correct for the total governmental domination of native populations in governments past. Tribal sovereignty is a complex set of laws disparate tribal laws as well as superseding federal governance.
What is the Bureau of Indian Affairs?
The Bureau of Indian affairs (or BIA) “[P]rovides services directly or through contracts, grants, or compacts to approximately 1.9 million American Indians and Alaska Natives. There are 567 federally recognized American Indian and Alaska Native tribes in the United States.”
There is so much more to Native American sovereignty than the fact that casinos are allowed on many Native American lands due to a lack of federal jurisdiction to regulate the practice. It is actually more than this though, a semi-sweetheart deal wherein the federal government eschewed reparations or greater grants of Native American (or even recognizing earlier, ignored treaties…) by the Reagan administration in 1988. A proposition was passed allowing gambling operations on reservations. This was much more significant at a time when Vegas was practically one of the only other American locations one could gamble legally.