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.
But, I'm going to prove to you that a school is the public square.
Students in any public school in America can wear jewlery, t-shirts or any other acceptable clothing advertising their religious convictions. They can pray in the hallway, they can proselytise between class and they can sing hymns under the flag pole if they so desire. All of these are speech Ron and it can be Constitutionally exercised in the public square, ie; schools.
Do you think the state can tell me that I can't wear my crucifx around my neck in federal buildings?