Is it possible to link the idea of the American Dream with any Puritan writer or with the Puritan era?
The American Dream has changed in its perceived definition and scope over time. It most certainly began with the formation of America and the ideals of the Puritans, among others. At its core, the American Dream is the concept that people should be free to live and worship as they see fit. America provided the opportunity for people to do just that. The Puritan settlers' American Dream was, in the famous words of John Winthrop, for their new nation to be a "city upon a hill," an example to all other nations for what they themselves could be. Although the Puritans were overzealous in their efforts and often, paradoxically, tried to force other settlers to conform to their own ideas of what freedom should look like, their intent was truly to make the world a better place in which anyone could achieve his or her dreams. Puritan-era writers of interest include the poet Anne Bradstreet and the great evangelist Jonathan Edwards.
Yes it is possible
Puritanism and Protestant Christianity had great influence on the culture and values of early Americans, so yes, Puritanism affected the idea of the American Dream. Puritans believed strongly in the concept of self-improvement, which to them stemmed from a desire to please God. This of course connects to the idea of the self-made man that is integral to the concept of the American Dream. Puritan values can be traced to the current ideal of American life- they were hardworking, pious, and believed in democracy. However, if one wished to criticize the American Dream, and challenge the idea of America as a meritocracy, he could look to Puritanism as well. Perhaps a darker side to Puritanism was their extension of the idea of predetermination: God punishes the weak, so people of a lower social class are being punished (and deserve their lower status). Echoes of this mindset are evident today, as some allege that those who are at the bottom of the social hierarchy (those for whom the American Dream has failed) are simply not hard workers. So, the Puritannical and American emphasis on hard work can have a darker side.