chickens sleeping in nesting boxes - advice needed

Hi
My name is Phil. I've been using this site as a point of reference over the last few weeks as I have recently become the proud owner of both an allotment and 3 hens.
The hens have settled in really well into their home with just one problem that  I wanted advice on.
They have both a sleeping area with roost and a nesting box. THe roosting area has lats on the bottom - so i can't put shavings down as they fall through the gaps. THe nesting box though is solid , I put shavings in there. The problem is that all 3 birds sleep in the nesting box, pooing everywhere, laying in their own mess, which , whilst no expert, I would anticipate would be bad for them long term.
Would it work to block off the nesting boxes for a few nights, enforcing them to roost?
The place where I bought them didn't have roosts, they were well cared for but in large dog run type housing indoors.
I hope someone can offer advice on this .
Thanks
Phil

Hi
Thanks for the advice
The coup/nesting area is mouse/rat proof, but I wouldn't say the run is.
Phil