I'd lay odds that it's the asparagus. Check out this site:
http://www.ces.ncsu.edu/depts/hort/consumer/poison/Asparof.htm
I know what you are saying with hiding in amongst the plants with a canopy over them. Mine don't free range but I did move into their run a bird of paradise in a great big planter (taller than they are). I figured those leaves are too hard to chew through anyway. So they like to sit next to the planters, in between them and the leaves provide shelter. Something like that could be a temporary solution until you figure out what plant to use.
http://www.ces.ncsu.edu/depts/hort/consumer/poison/Asparof.htm
I know what you are saying with hiding in amongst the plants with a canopy over them. Mine don't free range but I did move into their run a bird of paradise in a great big planter (taller than they are). I figured those leaves are too hard to chew through anyway. So they like to sit next to the planters, in between them and the leaves provide shelter. Something like that could be a temporary solution until you figure out what plant to use.