Timestamped link: https://youtu.be/n_NzXFs3Rqg?t=1079
On the show they had a guest which was a travelling preacher. A part of the questioning was about schools. Dorothy Kilgallen mentions, after the 'line' was revealed, if those are public schools, "I shouldn't think it'd be allowed now because off [illegibile]". She was cut off by the host before anything can be made more of it.
What was the issue, and how does this fit into the generalised topic of religion in schools during that period in the US and in the overarching timeline? The closest link i could find was https://en.wikipedia.org/wiki/Engel_v._Vitale about school prayers but i'm sure that's entirely past the mark.
The recent decision regarding prescribed prayer in schools is indeed the "can of worms" being referenced. Such prayers were quite a widespread practice, and the 1962 Supreme Court decision forbidding them was seen as a big change being imposed on society. Within a few years, public outrage had largely died down, and only jokes of the type "anyone who thinks prayer has been removed from the schools hasn't been in algebra class when a pop quiz is announced" remained. Beginning in the 1970s, evangelicals and right-wing politicians have tried to use prayer in public schools as a wedge issue, introducing mandatory "silent meditation or prayer" periods, defending prayers offered at athletic events or graduation ceremonies, and encouraging "meet me at the flagpole" student prayer groups.