Fixed it on Amazon Fire HD 10: go to apps, select Evernote; force quit; clear cache; clear storage. Launch Evernote and enter login credentials. I'm in. BTW, Evernote version from Amazon app store is 7.12.
Just got a new Amazon Fire HD10, and Evernote was the first app I installed. And I get this silly error too, "Login Error You cannot sign in at the moment. This is likely caused by a bad connection to the internet." Come on, Evernote, you can do better than this. I'm a premium user, have used Evernote since 2010. This bug is clearly affecting many people on a variety of platforms. Step it up, please.