That seems to be the big question. Here's the answer. Support freedom and stand for our values.
That may be harder than it seems. First of all, who in Egypt is fighting for freedom? What do they mean by freedom? Is it the same thing we mean, when we discuss freedom here in the US? I don't think it is. Freedom here includes freedom of speech and religion and a number of other things that would be unthinkable to the followers of Islam. Since Islam is the majority religion in Egypt, it seems likely then, that the people in Egypt, demonstrating for "freedom", do not mean that same thing by it that Americans would.
Secondly, what are our values? Christian values are criticized left and right. We are told that the founders of this nation were not Christians at all, but by and large deists. (A deist is someone who believes there is a god or gods, but who does not subscribe to any particular religion.) The fact that this is untrue, as can be seen by a cursory study of the founders writings, doesn't seem to matter, since the lie is now so well known.
(The truth, in my opinion, is that unless we once again embrace the values of Christianity, we are lost as a nation. But that's not the point of this post.)
If we say, for the sake of argument, that our values are Christian values, then we have somewhere to go with Egypt. Just as individuals will behave differently around those who espouse certain standards of behavior, so too will nations.
There is nothing wrong with telling Egypt or the world that we support the kind of freedom that we want for ourselves and that we will act according to our values. If they don't want to deal with us under those terms, then we won't deal with whatever government ends up in power.
The biggest problem here is the question raised above: What are our values? Without a concrete set of consistent values, we have no place to stand in the world. We will constantly be defending ourselves from propaganda and hypocrisy charges. That's a reasonable response to an individual or a nation that is not consistent.
We have become a nation also of moral relativity. This gives us no place to stand. It allows us to look the other way when someone cheats us or is rude or a bully. Christian values demand that we confront such behavior within the law.
This is one of the things that enrages so many liberals/progressives about our nation. Perhaps because they believe that by standing on those values, we are de facto conquering the world. Where are the conservatives to speak up and tell them that it is within the rights of those sovereign nations to change to our system if they want to?
The Christian values that formed this nation have been so wildly successful that, of course, other nations will want to adopt them for themselves. Why wouldn't they? And who are we to insist that they not?
If the people of Egypt want to adopt Christian values and the same kind of freedom that we expect here, then we should welcome them as a trading partner in the world with open arms. If they want to adopt a more repressive Islamist government, then we should restrict our dealings with them. That doesn't mean we need to do anything beyond communicating our intentions to the world.
Of course, that would be seen as nation building and bullying by the progressives/liberals. But they would be wrong. It is no more bullying to avoid those of unpleasant character in one's daily life than it is character building to let those same unpleasant individuals abuse you because you have no rational reason to deny them. After all, that may be what they see as good behavior, who are you to dispute it?
Perhaps we cannot expect nations to behave as individuals. Perhaps my analysis is quite incorrect and we should be busily interfering in Egypt. Promoting this individual and undermining that one. In my opinion, that is not consistent with the values of this nation. But then, I tend to think the values of this nation are and should be Christian values, even if you are not a Christian yourself.
What kind of hypocrisy are we practicing when we insist we want Egypt or any other nation to adopt one form of government over another when we, ourselves refuse to adhere to our own founding standards? When other nations look to us for the example of how a nation of free and independent people ought to conduct themselves and all we do is shrug our shoulders and say, "I dunno, does it feel right? Then do it." That's not how this nation became great and we do ourselves and the world a disservice by allowing it to be the case on our watch.
"And I will raise me up a faithful priest, that shall do according to that which is in mine heart and in my mind: and I will build him a sure house; and he shall walk before mine anointed for ever. "~ 1 Sam 2:35