Broodiness alone won't cause illness. However, broody hens eat less, drink less, and sometimes expose themselves to the elements, and these things can combine to weaken their resistance to infection and disease. This may be what happened with your hen. The good news is that she's isolated, so you can give her a treatment that's indidivualized (in case there is no sign of illness in your flock). And the light is a good idea for injured and sick poultry, because it enables them to burn their calories on other body functions (like the immune response) rather than for warmth. It also encourages thirst, and since most meds are water-soluable, this is a good thing.
I would not be too concerned about the eggs as the cause. Of course, you don't know the health of the flock the eggs came from, but generally, respiratory infections tend to come from stuff that chickens breathe in, just like our respiratory infections do. What's more, chickens are lower to the ground than us, and they dig in it with their claws, which can stir up bacteria & make it airborne, and then they put their face right down in the scratched-up earth and breathe in. And then there is dust-bathing! Because so many bacteria live in common soil, it it uber-important to keep your chickens healthy so they can fight off potential infections. Broodies are at a big disadvantage immune-wise. If you've got something in your soil that your chickens are kicking up - and ALL OF US DO - then this may be the reason she's the sicky and no one else is ill.
So, I'd recommend a good broad-spectrum antibiotic. If she's a favorite bird and you can manage it, I'd take her to the vet and have her examined, have a throat culture done, and get Rx that is designed to fight that particular microbe. But I am a crazy person (and strangely, I live w/in an hour's drive of THREE vets who treat chickens, which means I live in the Bermuda Triangle). If she is not a fave bird, or if the vet route is not possible, you still have a lot of options. But you will have to act fast. If she sounds like an old coffee percolator when she breathes, with a gurgling sound, then it might be bacterial pneumonia (very common & responds well to most antibiotics); it might also mean the air sacs are infected as well, which sometimes happens. It could also be CRD, or Chronic Respiratory Disease. You can 1) buy the water-soluable antibiotic that looks like it will tackle the infection & just give it to your girl; or 2) buy injectible antibiotic if you are comfortable sticking your chicken in the muscle or under the skin, if need be, and if you can get the needles & syringe for it; 3)you can try contacting Dr. Brown at First State Vet. I think you can call or email (if I were you I would call), and he can probably give you much more specific advise than I (he also sells medicine); or 4) you can search the board for respiratory ailment information (I am sure there are threads on this!), and see if there are more possibilities. NOTE - if you buy the antibiotics yourself at the farm/feed store, you may have to seek info on dosage from the board, as the packets all seem to be for like 100 birds, not 1. If using water-soluable, make sure you make a batch fresh every day & that your hen can get to it. A local vet or First State might be able to give you antibiotic pills, which you just kind of force her to take - this might be a good option since as a broody, her feed and water intake will be greatly reduced.
I hope this was helpful to you, and I wish you luck!
