I was able to resolve this.
I have No-Script installed and I allowed zen desk related stuff (zdassets etc). Not sure if that was related to Evernote or Evernote web clipper though, or just the related to the pages I currently had open.
The other thing I did was allow third party cookies, then checked all the Evernote-related hosts. (Note the first one starts with a . ) Then I saved those changes, and re-enabled Blocking Third-Party Cookies.