Small white things can be louse eggs or fly eggs. If she has been lethargic, the flies can take advantage and lay eggs on the living bird. You can check for lice especially lifting the wings and looking in the armpits. They are small and fast. A puppy/kitten safe flea powder is safe in the ill hen if there are lice.
You can examine her thoroughly, including opening the mouth and having a good look in there for mucus, redness, masses or anything else unusual. You can take her temperature gently with a rectal thermometer. Anything above 105F/40C is significant. Palpate the tummy for an egg, fluid, lumps or anything else.
She needs to see an avian/poultry-experienced veterinarian ASAP for complete examination, diagnosis and appropriate treatment. Check the link http://www.aav.org/association/index.php?content=activeMembersList
for members of AAV in your area or call your regular vet and see who they recommend. Many states/governments have poultry diagnostic labs that charge very reasonable fees to test for common diseases. Because you have others and presumably use the eggs, it is important to have a solid diagnosis and treatment safe for egg consumption.
She were my patient, I would start with complete fecal analysis and direct smear, for multiple parasites; bacterial culture and sensitivity of the feces and choana. Depending on the case I might do a fungal culture. Routine blood work is necessary to rule out other issues. Generally I start them out on antibiotics as indicated by the tests (I use a lot of human antibiotics that are injectable). I would very likely order a number of DNA tests for poultry viruses as well.
Pet/feed store medications and home remedies are harmful, ineffective, immuno-suppressive, and make them much worse and may interfere with the veterinarian's diagnosis and treatment. Do not use them.
Move the bird indoors to an aquarium, box or carrier with soft towels in the bottom, no perch, and food and water in low bowls that she can reach easily. Put the whole thing on a heating pad on low or medium. Keep her partially covered, warm and quiet.
Do not try to force food or water. You can offer warm cooked rice, pancakes, cornbread, grapes, melon, greens in addition to normal food. Transport as soon as possible.
She may need injectable antibiotics, calcium and many other medications. Act quickly and good luck.
Here are some sites of general interest:http://poultrykeeper.com/poultry-vets-uk/poultry-veterinary-practices-services-uk/http://www.thepoultrysite.com/about/http://mypetchicken.com/backyard-chickens/chicken-care/guide-toc.aspx