Does Anne Bradstreet's poetry reflect Puritan thinking?

Yes, it does, and quite to a large extent. For example, in her poem "Before the Birth of One of Her Children", she believes that God will grant her husband with a new wife if she dies which shows how much Puritan women believed in marriage and how God provided them with this gift. Her works talk a lot about the Puritan society, lifestyle, and the position that women had in such a society, as well as religion.