Well, define "Europeans." Although the Wikipedia article refers to the first Christian missionaries arriving in the 840s (which would make the Nestorians and probably Persians or Turks),
this article references it arriving in the 4th Century and I remember some new information indicating the first Christian missionaries might've arrived in the first century (!).
The earlier you get, the more likely the missionaries are going to be of Roman, not Persian, origin, since the first Christians were within the Roman borders (mostly).
Here's an article written by the PCA about the Nestorian missions to Asia.
http://www.aina.org/books/bftc/bftc.htm
It's got some anti-Catholic commentary, but it's really interesting. It's a good source of information on the Church of the East.
About the OP, the only idea I can think of is that the Mongols conquer Japan or the Philippines and Nestorian Christians get established there. The fall of the Mongol regime leads to persecution and an exodus in search of a new homeland that takes them into Alaska and ultimately the Pacific Northwest.
Alternatively, the article references the probable conversion of a Japanese empress to Christianity in 724. If there's a significant Nestorian presence in Japan, the behavior of the Mongols (who had prominent Nestorians) could cause them problems and they leave Japan in search of a new home.