What details in "The Devil and Tom Walker" reveal that Irving was specifically critical of the values held by the Puritans of Boston?

The narrator makes a rather snide allusion to the Puritan religious revival movement of the early 18th century, the Great Awakening, when he describes the setting as "the time that earthquakes were prevalent in New England, and shook many tall sinners down upon their knees."
The incursion of the Puritans into Native American territory is referenced from the Puritan point of view as "the stories handed down from the times of the Indian wars, when it was asserted that the savages held incantations here and made sacrifices to the Evil Spirit." Clearly, the religious beliefs and rituals of Native Americans was denigrated by Puritans, and the term "savages" is repugnantly Eurocentric.
The judgmental attitude toward others that Puritans allegedly held is implied in the devil's remark that "Deacon Peabody be damned . . . if he does not look more to his own sins and less to those of his neighbors." It is also noted that "Deacon Peabody, an eminent man who had waxed wealthy by driving shrewd bargains with the Indians." A Puritan ethos was that people should lay up treasures in heaven rather than focus on acquiring earthly wealth. And the word "shrewd" implies that he took advantage of the Indians, hardly a Christian behavior.

When Tom initially meets the devil, Old Scratch tells him to get off of his grounds. Tom responds by saying that the grounds belong to Deacon Peabody. The devil then comments,

"As I flatter myself he will be, if he does not look more to his own sins and less to his neighbor's. Look yonder, and see how Deacon Peabody is faring" (Irving 4).

Tom then turns his head and notices that the tree looks "fair and flourishing" on the outside, but is rotten to the core. Irving is addressing and critiquing how Puritans value external appearances more than a person's internal substance and integrity. Interestingly, Irving criticizes an ordained minister for being superficial.
Later on in the story, Irving critiques Puritan values by addressing how the church members react to Tom's zealous attitude towards religion. He writes,

Indeed, one might always tell when he had sinned most during the week, by the clamor of his Sunday devotion. The quiet Christians who had been modestly and steadfastly traveling Zionward, were struck with self-reproach at seeing themselves so suddenly outstripped in their career by this new-made convert (Irving 11).

Irving is critical of how a person's private sins are public in a Puritan community. Essentially, Irving is critical of the Puritan's obsession with external appearances and portrays them as a superficial group of people throughout his short story.