I wanted to do a bit of resarch first, I know that cats can get a condition called 'fatty liver' (hepatic lipidosis) from not eating. Apparently birds can get fatty liver, but theirs is actaully from eating a high energy diet. Looking at pics of bird livers with fatty liver, I would say that we can rule that one out. Blackhead seems the most likely culprit. It might be a good idea to worm all of your birds and follow a strict schedule. I have heard that it helps to work lime into the soil where you keep the birds, but I am not sure what that does... maybe discourage the earthworms? Earth worms are a vector for the parasite.
This morning the two remaining peachicks were both standing up at the feeder eating like little pigs! It was the first time I saw them eat in days. Also, in the past few days they have been lying down when I look at them.
If they are cured of Blackhead, will they become reinfected later or will they be immune for life?
I don't think they develope an immunity... blackhead isn't a disease, it's a protozoan. Though I have read that it affects the younger birds more. You'll need to worm all of your birds, and put them on a regular schedule to keep the cecal worms down. I am glad that they are actibng better!