sorry for posting in English, I get bored from the arabic keyboard. It is like every arabic thing, stupid.
Anyways, as for the religious assylum, I have no experience with it. I don't know whether the government in Egypt get to know about that or not.
Don't focus on the US and consider other countries if they are the same to you. You might postpone the decision to immigrate for a year or so where you start a serious process of applying for immigration to Canada, Australia and New Zealand. Of course it is premature to discuss that, but please consider it. Australia and New Zealand provide better conditions for new immigrants. Just consider it.
You might come here to the US and work illegally, but I definitly don't recommend it. It was easier before they made strict immigration rules . Some people make it through, some fall short.

As for the kids, come on, they will live in an healthy environment, both physically, and mentally. It is a hard decision, but even if you don't reach the same position as in EGypt, it is enough that you did your primary and most important task as a father the best way you can.
I know what egyptians say, specially if they have girls, and how the west is corrupted. Man, everywehere is corruption, but I saw here in the US more conservative christian girls than in Egypt, and in general ethics and moral standards are higher. Believe me, if you teach them Jesus at home and let them participate in the church, an orthodox church preferably but not necessary, they will be OK. At least there won't be any hypocrisy like what all islamic shitty crapy socities are built upon.

God bless you.