Actually, a rock was found in Mystery Mountain, Los Lunas, New Mexico that has the Aseres Hadibros inscribed in an ancient Hebrew script. The interesting thing is that the Hebrew script that was used is from before Churban Bayis Rishon.
We know from Divrei Hayomim that Shlomo HaMelech had a navy that traveled from Tarshish – which had access to the Atlantic – on a 3 year journey and returned with gold, silver, ivory, apes and peacocks. So it is entirely possible that they traveled to the Americas.
Of course, this is assuming that all the land masses were not connected at the time, (forming a giant super-continent known as Pangea) and they just walked over.
As to the Native Americans descending from Jews, I recall reading somewhere that the Cherokee Nation has an oral tradition that they were descended from some Jews who escaped Masada during the Roman siege. They believe in a very similar version to our Brias HaOlam ie. the world was created in 7 days by one creator and the first Cherokee Woman was bitten by an evil snake and brought death to the world.
More recently, I read somewhere that their DNA markers show that they come from the Middle East region.
Not sure what any of this proves, or if it matters, but it is interesting.