The philosopher George Moore describes basing "ought" statements on "is" statements as a "naturalistic fallacy." In other words, although one can judge the feasibility of ideal and whether a system of ideals is coherent with itself, one cannot real find a secure standpoint from which to judge them. Whether you believe religion has a role in government depends on two things. First, most atheists or agnostics would want government and religion to be separate. On the other hand, if you believe a certain religion to be true, you might wish that to be part of the system of government of your country, assuming that the religion itself is one that advocates a role for religion in government; some religions take this position while other believe that as government is secular by nature, religions would lose their purity and ideals by participating in worldly affairs.
Both The Scarlet Letter and The Crucible can be used to argue against religion having a role in government because of the possibilities of religious beliefs being used as a cover masquerading for personal jealousies or prejudices. Miller, in particular, is using the Salem witch trials as a stand-in for the anti-communist hysteria of the 1950s, and strongly disapproves of government enforcement of ideological or religious orthodoxies. Hawthorne's own religious beliefs are a matter of scholarly debate, although he was raised a Unitarian and was probably centered in a liberal Protestant tradition that advocated separation of church and state. The Scarlet Letter contrasts moral goodness with outward conformity and thus could be said to make a case against a legalistic national church.
Edwards was part of a movement called the Great Awakening that grounded the religious impulse in intense individual emotion rather than the formal strictures of a state church and thus would have seen state religion as leading to nominal rather than true belief.
However, even though these three works tend to advocate a separation between religion and government, there is no reason you need to agree with their positions. If you are a follower of a religion that believes that a divine being will punish an entire nation if people in it disobey certain religious laws, for example, you could make a coherent argument for the need for a state religion.