Modern Quakers indeed sing this song. 19th century Quakers may have sung apart from there meetings. So far as I know there is no evidence that there is any Quaker background to the song. Pete Seeger's source singer was not a Quaker.

There is no question that Robert Lowry wrote the tune & no one is sure of where he got the words from or if he wrote them himself.

Robert Lowry also wrote the music for lots of evangelical hymns. Here's more information about him. Marching to Zion is probably his best known.

And to quote Sandy Paton from the thread referenced above:
Doris Plenn was not a Quaker. She learned the hymn she later re-wrote as an Episcopalian. You will note that I hedged my bets in my earlier post by saying "I'm told" that Quaker meetings are traditionally silent (until someone is moved to speak, that is). The contemporary songs recently compiled into a modern Quaker "hymnal" (including the Sydney Carter contributions) would be likely to include "How Can I Keep from Singing" as a powerful song popularized by contemporary "folk" singers, along with similar songs of "spirit" and social awareness. To call it "a Quaker hymn" is to suggest that the later "adoption of an adaptation" by a group replaces the actual origin, which it surely does not. Boy, I must have made my position as clear as mud, as my mother used to say.

Sandy