My grandfather was born in the village of Mömlingen, Germany in 1907, and emigrated to the United States in the early 1930s. Here's his dad, my great-great-grandfather, and the city crest of Mömlingen:
You might note the two 6-pointed starsin German, the "zwei sechsstrahligen weißen Sternen". These six-pointed stars are related to the House of Erbach. The Erbachs were counts in the region. Here's their coat of arms, along with a detail from the tomb of Dietrich Schenk von Erbach, Archbishop of Mainz from 1434-1459. (Image via Wikimedia Commons)
Mömlingen is in the Miltenberg district of Bavaria, and just east is Odenwaldkreis, in Hesse.
No sechsstrahligen Sternen in Miltenberg. Instead, we get Bavaria on the top, the wheel of the Archbishopric of Mainz on the lower left and the "Franconian Rake" of the Prince-Bishopric of Würzburg on the lower right. Miltenberg, including Mömlingen, has been part of Bavaria since 1814, although it's just on the border with Hesse.
In Odenwaldkreis, we see some Erbach stars. I'm assuming, then, that based on the coat of arms of Mömlingen, (adopted in 1955, mind you) that it passed between Erbach and Mainz at some point in the past.
I've done a little digging on the Erbach family to see where those six-pointed stars came from, and how long they've been around. It looks like the Erbachs weren't particular as to how many points appeared on their coat of arms until about the 1400s, when they appear to have settled on 6. I've seen five and seven pointed stars in the pre-1400 period, mostly on cenotaphs in the Protestant church of Michelstadt. One very helpful tool is the really interesting German Inscriptions Project.
So far, there's no evidence that the Erbachs or their stars are related to the six-pointed stars of the Chicago flag. I doubt any will turn up.