I haven’t seen the statement, but the press coverage implies its authenticity.
When I worked at the Vatican, there was one morning when a Vatican employee had called into a radio show, identified himself as a Vatican employee, and then went on to talk about his relations with his live-in girlfriend. He didn’t use his name; so he wasn’t tracked down. But he certainly created a lot of buzz in Vatican offices. Everyone – even if they didn’t practice their faith much – knew that he’d violated one of the agreements of working at the Vatican. It’s like any business or organization, a certain amount of loyalty to the entity’s mission and identity is expected. When it comes to religious organizations, even more can be required.
Not every teacher at a Catholic school needs to be Catholic, but they do need to be able to interact with students in a way that upholds Catholic teachings. If they cannot do that in good conscience, then that teacher is not a good fit for a Catholic school.