It is assumed that modern humans and Neanderthals share a common ancestry. So, of course, they share DNA. In fact, humans and chimpanzees share 98.5% of their DNA. The logic behind claims that modern humans interbred with Neanderthals long after the two lineage split and after they both left Africa is as follows. DNA is found in non-Africans and Neanderthals that is not found in Africans. One explanation is interbreeding between modern humans and Neanderthals outside of Africa. Another possible explanation is that this DNA did once exist in Africans but became extinct as a result of genetic drift and that its extistence in modern humans and Neanderthals is a relic of their common ancestry. Same as with chimpanzees.

Now we find DNA (SLC16A11) common to modern people of pre-Columbian American origin and Neanderthals but apparently absent in Old World populations of modern humnas. If we borrow the logic used in the previous paragraph, we would conclude that modern humans interbred in the New World with Neanderthals. However, there were no Neanderthals in the New World. So we are forced to conclude that this "Neanderthal" DNA once existed in Old World mordern humans and became extinct as a result of genetic drift. We cannot do this without accepting the possibility that teh same thing happened in Africa with the supposed "Neanderthal" DNA found in non-Africans.

Never get scientific knowledge from the news media. there is a reason they majored in Journalism rather than science.