One of the best winter cover crops you can plant for clay soils is clover. Its a nitrogen fixer and helps break things up. Your birds will love it. Not so useful to you. Next best are various radishes. Good for breaking up soils, some feed value to you, your birds likely won't be nearly as interested. You can also grow winter peas, or if far enough north, fava beans.
Next year, you want to plant buckwheat, whenever best for your climate. Again, good for improving the soil, you can eat it, your birds can eat it. Consider the various squash as well.
Generally speaking, the fibrous to woody stems of most green growing things are of very limited nutritional value to your birds. Essentially, an afterthought. Needed nutrition, particularly proteins and many of the vitamins, tend to be heavily concentrated in the seed/grain/fruit, while the leaves contain much less, and the stalks typically even less than that. Excess fiber is not nearly so good for chickens as it is for us.
You can also consider applying lime/gypsum to your soil to help prepare it for use in producing vegetation. Check with your local Ag extension office, they may do free, or very low charge, soil testing.