My fiancée, her baby son, and I walked the Camino this summer and had no real problems telling locals and other peregrinos where we were from. Though we are not "patch people," we were openly carrying enough equipment, apparel, reading material, etc., of obvious American origin that the first guess of any discerning observer, especially after hearing us speaking, should have been that we were, in fact, Americans (or maybe Canadians, if they were willing to give us the benefit of the doubt). This was the case with most of our fellow peregrinos.
However, we found that most locals were stunned to learn that we were not Germans, which seems to be the default assumption for sets of blondish, blue-eyed peregrinos who aren't speaking French. It makes sense, both in terms of how many Germans are on the Camino versus how many norteamericanos they see walking to Santiago, and how American English (as opposed to British English, the distinctive phonology of which the locals are probably more used to hearing from pilgrims) sounds to the ears of people living in country where teaching English has only recently been made an educational priority. English and German are much more closely related to each other linguistically than either of them is to Spanish, so it's understandable that Spanish speakers who don't speak either of the other two languages might, through confirmation bias, mistake a dialect of English they aren't used to hearing for German.
The only major exception to this phenomenon that really sticks out in my mind was an elderly shopkeeper in Burgos who became very attached to the idea that we must be Finns and refused to believe us when we told him we actually hailed from los Estados Unidos.
In any case, we found that locals were usually enthusiastic to learn that we were Americans, except for a period of about two weeks after the US national team managed to defeat Spain's team in some sort of important soccer, er, fútbol match. We got along fine telling people we were Californians instead during that episode.
You must also realize that almost the entire length of the Camino Francés runs through what one might call the heartland of Spanish conservatism, and people you meet along the route are much less likely to be deeply offended by the United States government's relatively recent shenanigans (e.g., enticing Spain into helping invade Iraq in 2003) than people you might meet in other regions.
Historically, all the territory around the Camino Francés broke for the right-wing insurgents against the left-wing Republican government almost immediately after the insurgency began in 1936. You won't find much in the way of Spanish Civil War battle sites along the Camino because the whole region, surrounding the Camino like a sheath from the French border to the sea, was more or less uncontested Nationalist territory from the beginning of the conflict to the end, with General Mola making his headquarters in Pamplona and Franco making his in Burgos.
Demographics do shift over time, but with the exception of some areas around León, a peregrino walking the Camino Francés is still more likely to meet locals who voted in the 2004 general election (or in 2008, for that matter) for the right-wing Partido Popular, which sent Spanish troops to Iraq in support of the American-led invasion, than you are to meet locals who cast votes for the left-wing Partido Socialista Obrero Español, which campaigned in 2004, in part, on a promise to bring those troops home (and won).
In short, broadcasting the fact that you are an American shouldn't cause you much or any trouble with the locals over and above the usual stranger-in-a-strange-land issues that face anyone immersing themselves in a new culture. Just try not to be obnoxious about it, and represent your country well. :wink: