If you do not have a Facebook Account or you don’t want to use it in Hay Day, you can also connect using Apple’s Game Center. You may need to create an account for it, or login if you already have an account. This way you should be able to play the game without connecting to Facebook.
If you already connected to Game Center and still get the Facebook message when trying to access the roadside shop or newspaper, please make sure that you are receiving the "Welcome back, _username_" message from Game Center when launching the game (it should appear on the top of the screen). If you don’t, try launching the Game Center to see if you are actually logged in. You can also try logging out of the Game Center and then back in. Sometimes following some updates the log in is requested, even if you were logged in before the update. Completing the steps above should help you resolve that issue as well.