If you save your screenshots to Dropbox, use this Recipe to automatically add them to Evernote. Just make sure you configure the Dropbox folder name correctly!
If
New photo in your folder
This Trigger fires every time a photo is saved in the folder you specify. NOTE: will process up to 15 photos per polling period.
Then
Create image note from URL
This Action will create a new note with an image attachment in the notebook you specify.
Explore more great ways to automate Evernote and Dropbox