Thank you. I have never seen lambs quarters growing in any of our pastures, but it is good to be aware.
I am pretty sure that the innocent seeming Mystery plant, now that it has been identified as Sclerocarpus uniserialis, a member of the sunflower family, is actually the culprit. The extension agent said that all the members of the sunflower family can be toxic to the liver if eaten in large enough quantities. This plant grows (grew) heavily in the fence line of the pasture in which these two horses were living and in none of the other pastures. I needed to work on the fence so I pulled it all up and put it in the burn pile which was also in that pasture. The horses, who had never eaten it before, now considered it a treat and ate an unknown quantity of it. They fell ill shortly afterward. The timing of the events is what made me suspect the plant and want to identify it.
I am still awe struck by the effort you all put into helping me and am very impressed with your success. I also learned a lot. Many thanks!