The grey symbol on Amazing Grace hat looks like a Reichsadler (coat of arms of Nazi Germany)

It's a US Navy hat, not a Nazi hat. There's lots of eagles and hawks and whatnot in the patriotic regalia of many counties — in my personal opinion so much that I don't think we can reasonably just cede this one to the Nazis.

In this case, the hat is iconic — there are many wide-spread pictures of Rear Admiral Grace Hopper in dress uniform, and for me at least the association was automatic — especially with the horn-rimmed glasses and shoulder-length gray hair which are also part of the "hat" icon used.

Compare to this:

If you don't know who Grace Hopper is or why she is amazing, you should. If the association wasn't automatic for you, that's OK. Many of the Stack Exchange hats are obscure references or kind of "deep cuts". So here's an opportunity to learn — and this is something important that everyone who uses a computer or smart device should know.

A quick web search will find you a bunch of biographies, but here's some highlights:

• worked on the Mark I computer at Harvard during World War II, and wrote the manual for it
• put a literal dead bug in a logbook as a joke — this was not actually the first time the word "bug" was used, but the incident recorded due to her sense of humor is famous in the public consciousness. (I remember it as a question on the game show "Who Wants to be a Millionaire?")
• encouraged the practice of code reuse and code "libraries" in a time when that was not at all common practice
• invented the first compiler. This may seem obvious now, but definitely was not. It took her the best part of a decade to convince others of the utility and importance. Without her, it is very possible that computers would have stayed arcane calculators programmed by government scientists and specialized researchers.
• was instrumental in the development of the COBOL programming language, which I know is a joke for old fashioned today, but in its time was revolutionary in bringing computing to people who previously could not access it.

This isn't just some random American woman. It's someone you should know about. You literally would not be using computers in the way you do today without her.

And that's her hat.

You can learn more about Commodore Grace Hopper on wikipedia.

Here is what she looked like in uniform:

Hopefully this will clear up any questions on the origins of the symbol on the hat.

I think it looks more like the "Allies" faction's emblem from the Command & Conquer franchise:

A simplified icon can be interpreted to have a lot of different origins.

The difference between the Reichsadler and that icon isn't any less than the difference between the "Allies" emblem and that icon.

"the picture itself is still more Reichsadler than any other. See the straight line of the wings which is distinctive for this coat of arms." (source)

I disagree. Take these automotive brands, for example:

And that's just 2 of them.

Or how about United States Aviator Badges:

Since there's a few things stuck in comments, and I need to practice what I preach.

Since most of the hats this year are recycled, it might be worth it for our newer readers to take a look at the original hat. It had an even more obscure name - IDENTIFICATION DIVISION (yes allcaps), as an allusion to Cobol.

As for the less (or more Geeky Aspects?)

The Current hat is named Amazing Grace - an allusion to Rear Admiral Grace M Hopper. If you've not heard of her - do yourself a favour and read the wikipedia article. She was a pioneer, and helped found many concepts of modern computing

While not exactly accurate, the attempted renderation of the US Navy Hat is probably a lot more visible as a small addition to a avatar than the actual ones.

(warning. Boring pop history lesson follows)

On eagles - I'd blame western civilisation and their obsession with the romans. The United States styled itself on a idealised, and rather rose tinted view of the Roman Republic and many later empires styled themselves on the Roman Empire - say the french

Basically eagles everywhere.

And more or less, sometimes, a hat is just a hat.

This head and it's birdesque symbol does not even look like Nazi insignia. I mean at all. The head sticks out like a sore thumb, unlike the Reichsadlers all tugged in streamlinedness. Apart from what it actually is, it looks more like a Playmobil totem pole than a Reichsadler:

And yes, I'm old. I know the new one's look different and this old one is probably in some way culturally inappropriate.

And by the way, the heraldic symbol of Germany has been an Eagle before the Nazis, during their reign and afterwards. So even a German Eagle does not stand for Nazis, except very specific iconography. And there is no way to make it that specific in those... what? 20 pixels?