You should do some research to determine if it really IS in the public domain--and if it's not, do NOT approach the Girl Scouts yet. Proceed with the project but understand that if you are offered a contract by a publisher, or decide to self-publish, you will then have to approach them. Some kind of mutually beneficial arrangement might be possible. Or they might demand a share of the royalties.
Don't be too encouraged by Jan's experience--it's one thing to use a lyric as one small part of a novel, and quite another to build an entire text around it. But in any case, don't seek permission NOW.