Much of Hester Prynne's past in The Scarlet Letter is revealed through glimpses into the past that either she, or the character of Roger Chillingworth, bring up as the narrative progresses. Between these memories, and by analyzing the historical context of the novel, we can actually come to a conclusion as to who was Hester Prynne before coming to New England.
We know that the historical context of the novel is the mid- seventeenth century, in the settlement of Boston. This is colonial America, and the Great Puritan Migration of 1620-1640 coincides with the time period in which Hester is sent to Massachusetts, by her husband, from her native England. Taking this into consideration, we could conclude that Hester and Roger Chillingworth are both Puritans who migrate to America during this great time of migratory activity. At one point, Hester is even compared to Puritan leader Anne Bradstreet who, like Hester, also had to go through the harsh trip form England to make it in the colonies; this comparison may serve as further evidence of Hester's puritanism as the narrator is clear that both women have much in common.
Another possible indicator of Hester's Puritan roots come from her deep sense of loyalty to Chillingworth after he asks her to keep his identity private. Whether it is a reaction to guilt, the reality is that she keeps this very vital secret from Dimmesdale, perhaps because of an deeply-ingrained sense of loyalty to the figure of "the husband" that can only be acquired from her upbringing. This type of loyalty can be traced to Genesis 2:18, which is one of the most influential Bible verses in the Puritan faith. This verse reads
It is not good that the man should be alone; I will make him a helper fit for him.
Therefore, chances are that Hester, not by choice but by habit, felt the need to conceal the secret of her husband, despite of what went on between them.