Well if you do go the chick route, You should have a backup plan. like have the stuff for a brooder on hand. You could go with eggs, but your hen might decide that that is a great time to give up on being broody.
for your question about eggs making her sick- it would only make her sick if she never got up to eat.
you would also be aware that the other birds in your flock are not picking on the chicks.