No fake eggs will not make a hen go broody.
Once a hen goes broody in my flock I move her to an old cupboard that I place in the hen house. I have holes drilled in the door to allow a little light into it. I dust it with DE and make a nest in an old drawer in it and put the eggs I want to hatch in the nest and when the broody has started sitting tight, I move her into the cupboard nest and close the door. The darkness and the nest of eggs will usually mean she settles onto the new eggs. Once a day I open the cupboard door and let her out with the flock to eat and dust bath etc, then when she is ready to go back to the eggs, I make sure she gets back onto the correct nest and close the door. The first few days, she will go back to her old nest site and needs placing back in the cupboard but after that she finds her own way back to the correct nest. I usually put food and water in the cupboard with her so that it's available if she needs it.
The advantage of this system is that the broody gets a dark, quiet place to brood without disturbance from the other hens and they can't lay into her nest. I know she is safely on her nest when I'm not there and can't wander off and climb back onto a different nest of eggs by mistake. She still remains part of the flock as she eats with them once a day, so no reintegration problems and she can then rear the chicks within the flock. When they are hatching, I normally keep her in the cupboard for a couple of days and then I let her decide when she is ready to bring the chicks out. I have a large mixed flock including roosters and they free range every day. I've never lost a chick due to injury from another flock member and my broody hens are quite low in the pecking order. My first broody hatched 14/14 eggs and raised all 14 chicks without incident or loss. My Tasha is a total star though!
Anyway, that's the way I do it and it works very well for me.