I got an iPhone 6s for Christmas and with iOS 9.2 and Evernote Version 7.9.2, the sharing seems to be working now. (From news app with Fortune and Men's Health articles)
As a workaround, I would share through email. I set up a new contact with my Evernote email address. Named it Evernote Upload. I start typing that in the To: field and it pops up. Select it and send. On my iPhone 5s, this was faster than waiting on the Evernote share.
On the 6s, the Evernote share is just as fast as the email share.