Well, if I were you, I would refuse to quit. I would simply tell them that it is impossible for you to work the hours they are trying to give you, and that if they insist these are all the hours they now have available for you, they are going to have to terminate you.
Because if you quit because of the hours change, not only is it not illegal for them to do this to you, you will very likely not be eligible to draw unemployment benefits until you find another job. Because the hours and your not being able to find another baby sitter are considered personal reasons to quit the job. and that means very likely no unemployment insurance. Some states are more reasonable/lenient about personal circumstances for quitting and unemployment than others, but it's a gamble.
While it may not be at all convenient for you, please try very hard to exhaust all possibilities for finding baby sitting and accepting the new job schedule. I'm with BPrincess here, jobs are very hard to find, and even if you are approved for unemployment, you'd be receiving a lot less than you would working.
If it is a really difficult situation, could you work it out with a friend or family member so that you could do it for a short time, so that you would be able to begin looking for a new job with better hours while you are still employed?
Clearly discuss this situation (though you do not mention unemployment benefits) with your employer. Ask if they understand they are making it difficult for you to continue working by giving you the hours they know will be extremely hard for you. Try to negotiate. Do not threaten. They very likely know good and well that if they can get you to quit, there will not be a consideration of unemployment benefits. From what you said about the board and directors and such, they may be some kind of non-profit, and boy, those people HATE to have to fire an employee and get them drawing unemployment benefits. Don't believe them if they tell you, "Oh, we'll allow you to draw unemployment if you quit." Because they are not the ones who get to make the determination. If you are fired, they cannot stop you from drawing benefits unless they can demonstrate that they had a valid misconduct reason for terminating you. Being unable to comply with their sudden dramatic hours change after a long period of working one regular schedule is probably not going to be considered misconduct. It sort of looks like an attempt to force a quit on their part.
But even if they tell you it will "look better" on your job history, or references or whatever, if you go on and resign, do not do so. They cannot say whether or not you receive approval for unemployment benefits, so don't let them threaten you. But remember, a voluntary quit for personal reasons, even very excellent personal reasons is probably going to keep you from drawing benefits.
And in many states, (throwing this in because you are a single parent and might get in severe financial difficulty because you have quit your job and get no unemployment benefits) there are sanctions against you for quitting, and you cannot receive other types of aid like Food stamp coupons or TANF for several months if you've quit your last job.
