Flickr publish plug in has always worked faithfully until yesterday when it told me it could not contact the Flickr web service because it could not find an internet connection. Internet is up and Flickr is readily accessible directly. The LR Plug-in Manager says it is up and running and a re-installation here made no difference. Rebooting the computer also did not help.
As a follow-up, the help tab associated with LR will not connect to the internet either. It would appear there are some issues with LR not being able to see the internet all of a sudden
Re your problems with Flickr, many people have reported problems following Flickr's recent upgrade; see this topic in the official Adobe feedback forum: Problems with Lightroom 6 Publishing Services to Facebook and Flickr. Please continue the discussion there.