My siblings have chickens, but I wouldn't call them a money saver so much as a hobby. If anything the eggs are more useful for bartering/giving to friends, since even a family that eats a lot of eggs typically has trouble going through the eggs laid by a small flock. Take into account too that they're not going to be laying as much or at all during the late autumn and winter months. So you'll be drowning in eggs in the spring and summer, and then not really any. Historically people dealt with this by preserving eggs in lime (mineral, not fruit) water, but most people don't bother today since you can get them at the store.

You may still need to buy chickens down the line, because sometimes an illness can go through a flock and knock all (or most of them) down without there really being much to do about it. Letting them just breed for too long without introducing new blood can also cause incest issues. The vitamins and feed aren't really that expensive. You may be able to get a better deal at a local feed store than online or through a pet store.