Greetings, I am Dr. Pat. I have worked with birds for many years. I will do my best to help you.
First, ointments are oil-based and should not be used in birds.
Second, the cause absolutely MUST be determined so proper treatment can ensue. Conjunctivitis in finches can be caused by many bacterial infections, psittacosis, viral infections, spiral bacteria, fungal infections, contact, toxins and so on. They will have serious pneumonia and sinus infection BEFORE any ocular signs are noticeable, and those need to be addressed appropriately and aggressively.
For right now, just use artificial tears to soothe and lubricate. Very likely they will need prescription eye DROPS and systemic antibiotics.
Move any sick bird to an aquarium, box or carrier with soft towels in the bottom, no perch, and food and water in low bowls that can be reached easily. Put the whole thing on a heating pad on low or medium. Keep the unit partially covered, warm and quiet. Go to this link
for some ideas. White paper towels or white cloth towels will show the true color of the droppings.
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.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. They can be fatal.
You are going to need local help on this, and a scientific and solid diagnosis to find safe and effective treatment. It can be made worse by poor nutrition (seeds), seasonal changes, sleep patterns, cage cleanliness, etc. and can therefore be a bit complicated and require more than just a course of antibiotics.
You need to consult an avian-experienced veterinarian
ASAP for complete examination, diagnosis and appropriate treatment. Check this link
for members of AAV in your area or call your regular vet and see who they recommend; ask if they really have worked with birds a lot. If this 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).
Your birds may need injectable fluids, calcium, antibiotics and many other medications. Act quickly and good luck.
The following guidelines help with basic issues such as nutrition, obesity, good immune status. Surprising how the following can make a bird healthy, and how infrequently birds are ill if they are on the following regimen. No amount of medicine is going to work if the birds' basic needs are not met.
Finches should be on a high-quality, preferably prescription, pelleted diet (I prefer Harrison's High Potency
). In addition, they should be offered dark leafy greens, cooked sweet potatoes, yams, squash, pumpkin; entire (tops and bottoms) fresh carrots and so forth. No seeds (and that means a mix, or millet, or sprays, etc. etc.) and only healthy, low-fat high fiber people food. A dietary change should be closely monitored and supervised by your avian vet. convert to pelletsgood diet
Birds should get 12-14 hours dark, quiet, uninterrupted sleep at night. Any less and they can suffer from sleep deprivation and associated illnesses. They should be covered or their cage placed in a dark room that is not used after they go to bed. They should have access to bathing by daily shower, misting, bath bowl, etc. basic maintenance
The cage material should be cleaned everyday, and twice a day if the bird is really messy. Paper towels, newspaper, bath towels are ok. Never use corn cob, sawdust, wood chips, or walnut shell. Consider getting a large cage that is longer than tall--as birds move in a horizontal rather than vertical orientation; and have several feeding stations. cages
Never give grit, gravel sandpaper or cement perches. A bird will eat those to excess when it is not feeling well or if there is a nutritional deficiency. They do not need it at all (an old myth from the poultry days, even poultry do not need it). It can cause an impaction and lead to serious or fatal consequences. daily routinehazards