I don't know what happened, but your picture is too tiny to see and won't enlarge for me.
But I agree that just from what you said, it does sound like damping off. Damping off is a fungal disease, that leads to the base of the stem just above the soil level getting brownish and a bit constricted and eventually rotting so that the rest of the seedling gets cut off from the roots and so flops over and dies.
There is no cure for it, only prevention. Are these seedlings outdoors? For me, the only time I have ever seen damping off is my indoor seedlings and then only when they are pretty small. Once they get a reasonable size/ stem thickness, they aren't usually vulnerable to it. So if outdoors, are you putting very small seedlings out? You said they were from a greenhouse, so maybe they are starting to get damped off while in the greenhouse.
Greenhouse might be perfect conditions for it -- damping off usually only happens in conditions of too much moisture and humidity and not enough air circulation, as bill w suggested. Do you have a fan in your greenhouse to circulate air? What are conditions like in there?
PS after I posted this, I went back and now the picture is there and enlarges... I guess it just took awhile to load up. So now that I can see the picture, it doesn't look quite so much like damping off. Stem is brownish but seems pretty firm and not constricted. Is this just the beginning of the process? What do they look like before they die?
Anyway... whether or not, I am still thinking it is fungal and probably relating to too much moisture, whether from humiditiy, over watering, soil not draining well enough, watering the leaves instead of the soil, etc.
It helps if you tell us where you are. I presume in the Southern Hemisphere somewhere, where you are going in to summer now? You mentioned chicken manure. Is that well aged? Fresh chicken manure is too rich and can burn your plants.