Try going into Users and Groups under System Preferences, select your account name and seeing what log-in items you have. Is the Evernote daemon there? If not, open Activity Monitor in your Utilities folder and searching, using the search box in the top right-hand corner for "evernote". If you can't find anything running, try downloading the free EasyFind from the App Store and searching your drive for "evernote". Delete any files that you may find.
If Evernote doesn't come with an uninstaller and you just deleted the application, it might have (and sounds as if it did) left some extra files behind. I usually suggest that, if an application does not come with an uninstaller, people use AppDelete to remove the application and any associated files. I've never used Evernote and know nothing about it, but something is causing those messages to pop up and Evernote seems to be the culprit.
Are you using a mouse or the track pad? These pop-ups are context menus. Your RIGHT CLICK is stuck on. Go to Preferences >> Track Pad >> Point & Click and unckeck Secondary Click and see if it stops. If it is a mouse, then the right button is probably stuck (if there is one).