The circle of fifths concerns itself mainly with the relationships between the roots of chords or the tonic note of major/minor keys. Specific chords, like Cmaj7, are a separate issue.
In the case of a particular chord, the question is how it relates to the key of the music. In the key of C major, Cmaj7 is the tonic chord. You can find related roots of chords using the circle of fifths, but the the specific chord depends on whether you're remaining in the same key or shifting keys.
All of the chords in the given song have meaning within the key of C major. That is, they don't include any notes from outside the key. However, introduce a chord like D major (a fifth away from G), and now you've shifted the tonality away from C major, because the F# in the D major chord is not part of C major.
Study functional harmony to gain an understanding of the relationships between chords and how the circle of fifths fits into those relationships.