With 6 you'd only need 2 boxes, though since you've had issues with birds getting bullied out of them, maybe 3 would be better.
If you're sure you can get out every morning in time to let them get to the boxes and lay, I'd think it would be ok to put the boxes in the run if you prefer, since your run is fully enclosed. Nest box can be done pretty much any style, but I would make sure it's fairly well enclosed on top (your current ones are very open on top) and/or maybe put in curtains to add some privacy.
If you're sure you can get out every morning in time to let them get to the boxes and lay, I'd think it would be ok to put the boxes in the run if you prefer, since your run is fully enclosed. Nest box can be done pretty much any style, but I would make sure it's fairly well enclosed on top (your current ones are very open on top) and/or maybe put in curtains to add some privacy.