The fear expressed in this quote relates to a sense of "cultural life". The Puritans fled persecution and sought out a place where they could practice their religion outwardly as they pleased. The choice to establish a life in America was a choice of cultural life.
This group's religion and the practices that went along with it served to define the community. Survival of these practices and institutions was identical with survival in general, in the cultural thinking of the group.
Unity, therefore, was the best and only defense against the threat of "cultural death" wherein the institutions that gave the group its identity would be lost, eroded, or otherwise weakened or replaced.