When you try to perform an action like exporting to another app you will be presented with a list of apps that have declared that they can handle whichever "intent" WhatsApp uses for this. So if the problem is that EverNote isn't in the list it's not because there's something wrong with WhatsApp, but rather than EverNote hasn't declared itself as able to accept whatever export method WhatsApp uses here. And if that is the case there is nothing you can do to change that.
I don't use EverNote, but if I try to do this on my phone none of my note-taking apps offer to accept the export. Various chat and message apps (but not all), a couple of email apps, a couple of cloud storage apps, a couple of file managers and, implausibly, one ToDo app (the fact that an app declares it can handle a particular type of intent does not always mean it really can!).
The bottom line though is that I can really only see 2 options. You can contact the app developer, ask them to add the function, and see whether anything happens (and, in the unlikely event that anything does, how long that takes). Or you can try exporting to something else and then copying the data from there to the app you want to keep them in.