Sorry to break the intellectual aura of The General Discussion Drunkbar, but that was never in the Constitution, that's in one of Thomas Jefferson's personal letters...
While the exact phrase "separation of church and state" is indeed from Jefferson's letters and not the Constitution, the Constitution quite clearly expresses the exact same sentiment with "Congress shall make no law respecting an establishment of religion". This is further supported still by the No Religious Test Clause of Article VI.
Furthermore, I find it funny that people constantly try to bring up this whole "It was just Jefferson!" argument, when he was, you know, kind of involved in the creation of the Constitution and became the third President. I think it is safe to say that his external writings have some merit in determining what exactly the Founders meant.
Also:
This means that a government-mandated church (state church/ church-state; think Great Britain around that time and pretty much all of those years before said time) is not permitted or tolerated.
It protects religion (or the lack of it) from the government, not the other way around.
It's not freedom from religion (or the lack of it), it's freedom of it.
By definition the street must run both ways in order for stable freedom to exist. If one has freedom of religion without freedom from religion, then the government can and will establish religion in some form (even if this does not mean going to the extent of having a State Church). As this power is most likely to fall with the largest religious group, all minority groups will be oppressed. Meanwhile, if one has freedom from religion without freedom of religion, then the government can and will oppress any religious group that they dislike. As this power is most likely to fall with the largest religious group....all minority groups will be oppressed.
The United States holds a great honor in being the first secular state of the modern era, one that is too often forgotten or done a disservice by arguments such as your own.
Egypt isn't "separating church and state", there was never a joining. The church was, became the state. It "separated" every religion from the new Islamist state, except for it's own, of course.
There was most certainly a joining under the Muslim Brotherhood's administration. The church did not become the state. The church becoming the state would be the actual religious authorities having direct political power (such as in Iran), and the lack or depowering of a civilian administration in favor of a religious one. That did not happen. This, the exercising of religious oppression by the civilian administration, is just the result of not separating church and state.