The idea that America is a "Christian Nation" is important for the Christian Right. They need people to believe this in order to sell their agenda to enshrine their form of Christianity over law, politics, and culture. It would be nice if liberal Christians consistently opposed this rhetoric, but not all do. Hillary Clinton, for example, doesn't go quite so far as to use the phrase herself but she has been more than willing to support the idea that America is a nation for "people of faith." This excludes people who don't have religious faith in any gods, but why does Hillary Clinton not want to include secular atheists as equals in America?
In a statement denouncing a court decision striking the phrase "under God" from the Pledge of Allegiance, Hillary Clinton wrote:
Ours is a nation founded by people of faith. People of faith have helped lead some of the most significant movements of social justice throughout our history -- to end slavery, to win civil rights for all Americans
Hillary Clinton is making several implications here, all false and all bigoted. She is implying that religious ideology but not secular ideology was key to founding the nation and "movements of social justice," which is false. The American government is in fact founded on secular political ideology and a secular document, the Constitution. It was written by people with Christian and Deist beliefs, but they didn't think that their religious beliefs should play any role in the government itself. Clinton, though, is arguing the opposite.
Hillary Clinton is implying that people without her religious faith — atheists at the very least, but also perhaps non-Christians — were not important to "movements of social justice." That is as wrong as saying that gays have played no major role in America's social and political development. Hillary Clinton is also implying that religious believers and religious ideology were not important factors in those forces opposed to social justice, which is an outright lie.
What exactly does she mean by "people of faith"? This is a phrase created by the Christian Right to mirror the liberal term "people of color." It's supposed to give the impression that one is being "inclusive" of all religious faiths and only excluding dirty little atheists, but there is no evidence that it is meant to apply beyond Jews and Muslims at most; on the contrary, most uses of "people of faith" could simply be replaced with "Christians" and you'd be left with a speech or essay that looks about the same as what Christian Nationalists have been saying for decades.
Even if we pretend that "people of faith" really does refer to all theists, regardless of what religion they have and even if they have no religion, then at the very least it is a deliberate attempt to form a circle around theists generally and define atheists as outsiders. It's an attempt to say that theists have more in common with each other than they do with any atheists, that theists are "insiders" in American politics, culture, government, and society, and therefore that secular atheists - as outsiders - need not be treated as equals.
Even if Hillary Clinton doesn't say that America is a "Christian Nation," she is trying to say that America is a "People of Faith Nation," where at the very least atheists are outsiders. If we read "people of faith" as it probably should be read, which means excluding just about everyone but Christians and Jews, then we simply traditional "Christian Nation" bigotry that's been rephrased in a manner that will sound more palatable to liberal and moderate Christians.
...we should never forget the blessings of Divine Providence that undergird our nation.
Why does Hillary Clinton believe that America's liberties have anything to do with "Divine Providence"? That's the sort of rhetoric we should expect to hear from Christian Nationalists when they claim that America does well when it obeys God but suffers disasters when it disobeys God.
No one is required to have faith, and our government does not impose faith on its citizens.
That is true — it would be illegal for any level of American government to impose any faith on people. Hillary Clinton does, however, want our government to send the message that people who have faith in her god are politically and socially favored. If these "people of faith" cannot be have their faith imposed on the rest of us, they can at least ensure that everyone else is reminded that they are inferior. This is the purpose and effect not just of having "under God" in the Pledge of Allegiance, but of Hillary Clinton's entire defense for keeping that phrase.
Imagine if the term "people of faith" were replaced by variations on word "white" - would that not be recognized as blatant bigortry? What would be the purpose of proclaiming that America was "founded by whites" except to tell non-whites that they aren't as important, that they aren't equal, and that they should expect to be treated as second-class citizens?