Although there are some scholars who question whether Roger and Hester are actually Puritans, there is enough evidence to show that they hold at least one Puritan trait that they both share. This trait is their common belief in a "ruined soul."
The idea of a "doomed and ruined soul" is at the epicenter of the Puritan faith. It is precisely the fear of this impending doom that is used consistently in sermons which have historically shown how often this idea comes up. Moreover, this sense of doom is a good conduit to teach the otherwise obedient and feeble parishioners to be good and behave...or else.
Look at this stanza in The Day of Doom Stanzas by Michael Wigglesworth (1662). This book was so powerful that it became, literally, the first best selling book in American history. Now take a look at how it shows that ruined souls and their impending doom are a preoccupation in the Puritan community.
Before his Throne a Trump is blown,
Proclaiming th' Day of Doom:
Forthwith he cries, Ye Dead arise,
and unto Judgment come.
No sooner said, but 'tis obey'd;
Sepulchers open'd are:
Dead Bodies all rise at his call,
and's mighty power declare.
That being said, look in chapter IV, where Roger and Hester are making a promise to one another that they will not reveal the fact that they (although they have not really divorced or anything) were once married. The fact that Hester is true to her word places her in a submissive position where the husband is seen still as the head of the family, even though they severed ties.
However, it is the idea that Chillingworth will ruin her soul, that makes Hester fear him the most:
ROGER.."How is it, Hester? Doth thy sentence bind thee to wear the token in thy sleep? Art thou not afraid of nightmares and hideous dreams?"
HESTER..."Art thou like the Black Man that haunts the forest round about us? Hast thou enticed me into a bond that will prove the ruin of my soul?"
ROGER-"Not thy soul," he answered, with another smile. "No, not thine!"
Even though Chillingworth is a scholar who is also versed in natural sciences, he does not scoff at her mention of the "ruined soul." He actually acknowledges his own belief in that concept, and even says to her that it is not HER soul he plans to ruin, but Dimmesdale's (he does not know it is Dimmesdale yet). Therefore, this one trait of Puritanism is one thing that is shared in common between the former husband and wife.