Yes, your birds can have them !!! They prob do (this is the time of year for them) !!! You need to start checking them (like I said... pick them up and look under there wings and near there vents) Then if you see them (at this point it wont hurt even if you dont since they are on the eggs) Go to your local feed/
TSC and get some poultry dust for mites/lice and dust them all !! Then repeat dusting in 10 days to make sure they are all gone !! Make sure to get under the wings and around the vent VERY well !!! It is VERY important to get rid of them since your birds can get very sick/weak from the mites/lice if the bugs aren't taken care of !!
I have found a good way to dust is to get a $4 tote from
Walmart put the bird in it and dust in there, that way its not such a waste of dusting powder !! Also helps if there are 2 people doing it... Then you will need to clean and dust the coop ( I know you said they "free range" but they sleep in something right ? ) Another thing you can use for the coop dusting is DE (food grade) ask for it at your local feed store, just make sure its food grade.
As far as the eggs go.... Im not sure on how to get rid of them on there.... I had a hen sitting one time who had mites with eggs under her... I dusted her and the mites on the eggs died also... never had them in the bator... After this hatch i would bleach the bator and put it in the sun.
Hope someone else can give you more ideas on what to do with the eggs !! I guess at this point I might think about dusting the eggs and cleaning the bator since when the chicks hatch they will get the mites on them.