I wonder why the EU uses a Nazi flag like the one from this picture?
- That isn't a Nazi flag.
- The EU didn't use that flag.
and the Betsy Ross flag:
The picture is just another example of far-right extremists wrapping themselves in the flag of the country they purport to represent. In this case, history shows that they were not representative of the majority view in the United States at all.
As the article cited above observes:
In February 1939 Kuhn and the Bund held their largest rally in Madison Square Garden — ironically, one which marked the beginning of the end for the organization.
The German-born American leader of the German American Bund, Fritz Julius Kuhn, was deported from the United States back to Germany after the war.
It's perhaps worth noting that there is a flag bearing the symbol of the Bund between the two American flags in the picture linked in the question. The lack of resemblance between that flag and the EU flag is striking!
There doesn't appear to be any suggestion that the design of the EU flag was based on the design of the Betsy Ross flag. The circle as a symbol of unity is a common theme (see, for example, Squaring the Circle - Geometry in Art & Architecture, by Paul Calter).
As the Council of Europe described the design in 1955:
'Against the blue sky of the Western world, the stars represent the peoples of Europe in a circle, a symbol of unity. Their number shall be invariably set at twelve, the symbol of completeness and perfection'.
— Council of Europe. Paris, 7–9 December 1955