During the American Civil War there were ethnic regiments in the Union Army. Specifically quite a number of German-American units where the language of drill was German, at least one Norwegian-American unit from Wisconsin using Norwegian for drill, and various others which had to some extent a certain ethnic identity. This was to entice fresh immigrants in to the military. By WWII the number of "new" immigrants of military age was much smaller. Sure a lot of first or even second generation "hyphenated" Americans had a strong identification with their heritage, and also good to native language skills. OTOH as far back as TR there had been a big push to do away with "hyphenated" Americans under the rubric of we all are "Americans". This was pushed during WWI where there was a good bit of publicity of the Rainbow Division with a mix of men from many states and many ethnicities.
In 1940 the divisions were much less on national origin, or religion and much more on the basis of skin color for separation. Thus you had the separation of segregated black units and job categories. Where you had significant Japanese communities, like California and Hawaii you had Japanese in the National Guard units, especially a high number in Hawaii. Japanese-Americans were in ROTC in various universities - they were not separated out until after Pearl Harbor because of fears about divided loyalties. As others have noted, German refugees from Nazism, including Jews, were viewed with suspicion by US authorities when they attempted to join the military.
Refugees and various ethnic minorities were used by the US military more for specific skills rather than in "segregated" units. Yes some units, like Alaska Scouts, were very ethnic (in that case Inuit) but otherwise you did see Japanese used in the Pacific as language specialists, like wise others with language skills either in regular army detachments in Europe or with the OSS. In fact, any Americans with heritage from an Axis power tried very hard to NOT be identified by that and rather to be "American".