In order for Song Reader
to feel more authentic, Beck filled it out with fragments of songs and melodies, including fictional album collections and song titles. The back of "America, Here's My Boy" refers to an album of "Late Patriotic Song Successes!" all about soldiers and war and the like. (A popular genre of sheet music back in the early 1900s.)
The idea of "Late Patriotic Song Successes" comes from the back of a piece of 1918 sheet music which you can see here
. Many of those titles on page 4 are similar to the song fragments Beck wrote himself; and in some cases the lyrics are similar too (though Beck's melodies seem to be his own).
One of the 1918 patriotic songs on that link is called "When A Boy Says Good Bye To His Mother And She Gives Him To Uncle Sam."
Beck clearly adapted that into his own verse, flipping the point-of-view of the song. His lyrics and melody don't match it any way either.
Lastly, Beck's fragment is credited to "Georgia Larson." Most of the names in Song Reader seem to be either made-up, or references to people he works with or at McSweeney's, who published it. I could not find who Georgia Larson might be a reference too.