I don't know if Facebook ever had RSS support for Groups; Groups used to be pretty limited. They don't now.
Pages though, they've always had RSS feeds. It's just getting a lot harder to find.
I may be dreaming, but I thought there used to be an official RSS link somewhere. If so it quietly passed on, maybe in the last 6-12 months.
The old Feed URL format still works though:
https://www.facebook.com/feeds/page.php?id=121212where the number is the Page ID.
The trick is finding the Page ID. Facebook keeps moving it around. As of June 2015 find it here:
- Click About link beneath Page photo (Timeline, About, Photos, etc)
- Find Facebook Page ID at bottom of the About Link.