While I've read much, especially about the Marshalate, this is probably one of the toughest questions to answer. I'm personally a Ney Atheist, I don't see any credible evidence to him running towards the US for a few reasons.

First, language. While there is far more linguistic plurality in Early America, English isn't as common in Napoleonic France (or Europe) as it is now. For Ney to come to the US and speak French wouldn't be impossible for him but for him to speak English at a fluency that would be required for a teacher, it wouldn't be possible.

Ney was one of those highly prized officers which spoke German, being born on the border of France and Germany (his birthplace is now in Germany). As many of Napoleons Allied forces spoke German, German speaking French officers were of special importance. However, from my studies, I have not heard of him learning English.

Second, it's not his personality. Ney was... not the sharpest of the commanders. He was certainly an effective leader and a mixed quality commander, but he wasn't a particularly known bookworm like others such as Davout or St. Cyr. He was bound for the Civil service but was quickly bored of the life and joined the pre Revolutionary French Army, where he rose through the NCO ranks.

This goes further than sources as studying an individual or series of individuals within a group, you get a sense of their... personality, even of it is a highly debated thing as stories and knowledge is filtered through the lens of Napoleonic Glory and Admiration. There's a story in Delderfield's *Napoleon's Marshals" where upon reaching Poland after the disastrous retreat from Russia, Ney simply reported in to announce that the army crossed the Dniper and requested a bowl of soup. He did his job and just wanted to warm up, fleeing is something that he never did.

Third, this is nothing more than a wild story. This reaches the main issue within history, what is true? There is a significant lack of evidence to prove that Ney made it there and there is a very significant amount of evidence to prove that Ney was executed in France. How does one prove something that you can't definitively prove? When we look at such things, we deal more with probability, which is more likely? Ney took his execution and died? Or did Ney secretly learn English and decided that a quiet and bookish life was better than death?

Knowing what I know of Ney, the rear guard of Napoleon, the most brash of Napoleon's commanders, the Bravest of the Brave, I am sure that he would rather face death than paperwork. Remember, it is at Waterloo that Ney supposedly says "Come see how a Marshal of France dies." [Edited in] Considering Ney's actions at Waterloo, he would only go out in a way that would be proper for Ney.

Sources:

Napoleons Marshals By R. F. Delderfield

Napoleons Marshals Ed. By David G. Chandler

The Campaigns of Napoleon by David G. Chandler

Edit to fix things from typing on phone with an added sentence on the last paragraph.