"Euro-skepticism" and "states rights" are not equivalent, and therefore we would not expect there to be a similar effect. The closest equivalent in European terms to "states rights" would be "devolution" in the British context.
The United States is, at least since the beginning of the 20th-century, an uncontested country, state and nation. As such, questions surrounding issues of 'states rights' are not about competing sovereignties, but about the constitution of the nation wherein all parties see themselves. While states and the federal government have some tension over sovereign control, individuals participate—or at least can participate—in the selection of both governments. As such, while people may view themselves as Virginians, Texans or Californians their national identity is indubitably American, and that identity has not been in question for at least 100 years. (The Civil War was partly over this issue, but it is not a modern construct.)
Euro-skepticism, however, is about competing sovereignty, and more importantly competing sovereignty with an entity viewed as illegitimate and undemocratic. Euro-skeptics accept that they are geographically European, but believe, rightly, that the EU is not a sovereign state. Consequently, their national identity remains with their local state, of which they are a citizen.
In other words, 'states rights' in the U.S. is group internal competition over the structure of the state of which people already see themselves a part, whereas 'Euro-skepticism' is competition between an in-group—the state of which one is a citizen—and an out-group—Europe. This dynamic explains all other differences.
While states-rights advocates may be more or less pleased with the policy choices or internal structure of the U.S., they consistently see themselves a Americans. Did they not, they would not advocate states rights, but independence of their state or region—and there are separatists in some areas. Consequently, there is little change in imagery, or greater or less willingness to identify as American. Similarly, one could be 'chauvinistic' or not, regardless of ones position on states rights. The overlap between 'devolution' in Britain is helpful, because while the SNP may like devolution, they want independence, and consequently are 'Scottish', in contrast to Tories who favored devolution, but saw themselves as primarily 'British' even when they were in Scotland, England or Wales etc.
In Europe, on the other hand, the fact that Europe and other Europeans are the out-group makes use of European symbols and identification as 'Europeans' in a political sense completely strange. A Euro-skeptic in Britain (or France, or Italy etc.) is barely more a 'European' than she is a Pole, or a Spaniard. Consequently, the European political identification and symbology makes absolutely no sense whatsoever.