Let me ask you this Ron. If the teacher leads the class in a recitation of the Declaration of Independence which declares that our rights are granted by the Creator, is he violating the Establishment Clause of the Constitution?
Nope. The DOI says what it says. The teacher is teaching history. It would also not be a violation to discuss what is known about the religious beliefs of the writers of the DOI, and how it may have influenced what they wrote.
Now, if the teacher teaches whether or not those religious beliefs are right or wrong, then that would (IMNSHO) be a violation. Leading the students in any kind of religious exercise, or supervising the class while a student did so, would also be a violation.
A military base is only one example of a tax-supported institution that I don't consider "the public square" (besides which, to be a public square, I'd think that access to a location would not be limited to the military, but be open to the general public). The floor of the Congress is another; only elected officials can speak there. The inside of any Federal office building is yet another; you can't even go in there unless you are an employee, or are entering inside to conduct specific business related to that installation's function. There are numerous tax-supported institutions that, in my opinion, aren't properly described as "the public square", and it seems to me that the public schools are one such.
Now, it seems to me that public parks are "the public square", and the sidewalks in front of federal installations can be "the public square" if proper arrangements are made so as to not block traffic. These are places where one can reasonably expect to address the public with minimal interference. And if you instead choose to publicly address one's God, that's fine too. But I believe that your definition of the "public square" is over-broad.
I make a point of this because I've seen a few people defend government-led or -supervised group prayer in public schools on the basis of the schools being "the public square". It's a fine-sounding phrase, but upon reflection I thought to myself, "What do they mean?". I think of a guy on a soapbox, addressing passers by on whatever comes to mind, or a group holding a rally and making speeches on the issues of the day. So, when you used it, I wanted to see what you meant. I haven't seen a lot of logic applied to what people think the public square is, and what's reasonable to expect there.