I think it all depends on what type of job that you have. If the activities of your job are 'mission critical' then the issue becomes a bit sticky. There are some jobs where people have to perform duties that are crucial to the day-to-day operation of the company or are necessary to be filled to keep the public safe. On the other hand, my job (I'm a teacher) is one where someone can come in a substitute for me. Yeah, it's not the best solution but the world won't end, production won't come to a complete halt, and peoples' well-being won't be jeopardized.
Also, depending on the type of job/industry (public? private?), I can see how your boss might be hesitant to give anyone (not just you) a day off because of a 'religious' holiday. I'm not saying you're wrong and he's right or you're right and he's wrong, but if he gives you 'Good Friday' off then he should also be giving other employees Rosh Hashanah, Yom Kippur, and other religious holidays off. As a boss, he has to be fair to all employees and if he lets the Christians off for holidays, then he must let everyone else off for their religious holidays. I understand the situation you are put in, but this also puts him in a touchy situation as well. Although I wouldn't care one bit for my boss to find me a sitter that I knew NOTHING about, at least he gave some effort to try to reach a compromise. Albeit not a really great solution...
Have you tried to take a personal day? Vacation day? Comp time? Maybe try suggesting to him that you'll 'make up' the time but that it is critical that you take care of your children as they have nowhere else to go. I'd try to speak with him face to face, calmly, and explain your situation. If you've been an employee with an exemplary attendance record, point that out (such as, "I have always respected the need to be present at work and fully productive, and in my X number of years here, I've not taken advantage of the amount of leave time that's been extended to me. I can assure you that this is one of those special circumstances where I clearly have no other option but to not work."). I'd try negotiating and compromising instead of being combative and attacking your boss, still being assertive about the needs of your kids.