I would not use the favourite box to "encourage" a broody as it will cause problems with the other hens laying in her nest if it does help trigger broodiness. In your situation I would hang curtains across one of the least used nest boxes and put the golf balls in there. Don't be surprised if they all suddenly have a new favourite nest box though!... hens prefer a dark secretive, safe place to lay. Usually the favourite nest box will be the darkest one in a corner somewhere.
@ChickenCanoe
....I also remember reading that piece of info somewhere on BYC a few years ago and have never seen it referred to since. I think it may only apply to a hen that is already in the pre broody stages though. My best broody hen (she broods twice a year every year) and her daughter amassed a secret nest of 47 eggs (a veritable mountain of them ) a couple of years ago, but showed no inclination to incubate them during that time, so the pressure alone does not appear to be enough. It was Christmas holidays time though, so not really a natural time to raise chicks. Then I have 2 pekins (bantam cochins) at the moment that are determinedly broody with no eggs at all, so I think it is one of those factors that may work for some but not all. I suppose with hormones, as we all know, there are no hard and fast rules!