First of all, I should say that I have no expertise in genetics, but I do find the subject interesting and have done some reading.
Before you consider DNA tests , I think that you should be aware that analysis of Y-DNA would show only the origin of your gf's male Shaw ancestors; it would not tell you anything about the origin of his female ancestors. This test would confirm Native ancestry only if your gf was the direct male descendant of a Native man.
As I understand it, it is the mitochondrial (mt-DNA) in females that is used to trace the geographical origin of female ancestors. For this test you would need a female descendant of your gf's sister (if he had one and you knew the descendants). Men do inherit mt- DNA from their mothers, but they do not pass it on to their offspring. An analysis of the mt-DNA of a living male Shaw would reveal only the origin of his mother, her mother and so on, not that of his father's female ancestors.. If you do have a DNA sample from a male relative analysed and it comes back negative, you will never know for sure, without the female test.
If Mr.Shaw did have some Native North American ancestry, I would think that it most likely came from his mother, but if the Shaws had been in Canada for a generation or two, then perhaps someone on that side. I imagined the scenario as: Irish immigrant or son of same, married a Canadian woman who had some Native North American ancestry.
As Redroger said, cost varies. Here in Canada, it runs anywhere from about $100 - $600 or more for really complex analysis.
Cheers,
Carmela
.