This poem shows that it is definitely not written by a Puritan in a number of different ways. Key to understanding this point is focussing on the speaker's view of the afterlife and identifying how sharply it differs from traditional Christian notions of heaven and a celestial realm where we end up. According to the speaker of the poem, what happens when we die is that we re-join nature:
Earth, that nourished thee, shall claim
Thy growth, to be resolved to earth again,
And, lost each human trace, surrendering up
Thine individual being, shalt though go
To mix forever with the elements...
This view of the afterlife, where we just return to the dust from which we were made, thus stands in sharp contrast to Puritan notions of heaven and an afterlife spent with God.