The Evernote extension 7.29.2 {E0B8C461-F8FB-49b4-8373-FE32E9252800}.xpi has in manifest.json:
"incognito": "not_allowed",
so it is on purpose. I believe it should apply to all WebExtension browser but it does not in Google Chrome at least.
If anyone can get a clue as to why they prevent private clipping (while Firefox which as far as I know own Pocket allows clipping in private mode), please add it here.
If I remove this line from the manifest.json, repack the xpi (with the zip command), load the now not correctly signed extension (by setting temporiraly in about:conf xpinstall.signatures.required to false) I am able to clip in private mode with Evernote web clipper 7.29.2.