Hard to say. I'd seal them in the fridge and use the sniff test. I don't think I'd keep even a fresh hard boiled egg more than 2-3 days anyway, but that's just me, I don't really have a good reason for that cutoff.
My brother lived on a boat, and ate eggs that were over a month old.
No refrigerator on the sail boat.
I would think that if they are cooked they would be fine for the chicken.
As long as they don't smell rotten.
Weeks! You don't even want to know how old store bought eggs are, and they are often left in the back without refrigeration. you can give them the float test before you boil them.
If they sink right down, they're safe for human consumption.
If they lift up on one end but still rest on the bottom of the bowl/pitcher/whatever, they're still good but use them first, there's more air in them.
If they rise off the bottom but not too high, (say an inch or so) I feed them back to hens or dogs since I have a surplus, if I was 'hurting' for food or supplies, I'd still use them, maybe for baking etc.
If they rise say more than an inch or two, I wouldn't eat them but definitely still safe for chickens.
If they come to the top, I wouldn't eat them or feed them.
In the refrigerator I let mine stay at least 4 weeks before I boil them otherwise they're hard to peel. Most of the time when I collect eggs I bring them in and set them on the table and most of the time they're left there a day before I put them in the refrigerator.
My mom tells me that eggs that have gone bad will float like a buoy in water. They're so full of bad bacteria that the gasses are trapped and will cause it to float like that. Throw away the ones that float like that. You don't want to make your chickens sick.
I agree with portage girl on the float test. When I want to do potato salad I let them sit out on the counter trying to get them stale enough to make good hard boiled eggs. After a week sitting out they still are not as stale as "fresh" eggs from the store.
I think we would all be appalled at the age and handling of store eggs.