As an RN, I have had to care for a number of patients with back injuries and life long disabilities. I am very fortunate that I have not had any back problems, and I go out of my way to protect my back both on the job and at home. I hope you are getting proper treatment for your back condition. I know it can be difficult with back injuries, pain, and pain management.
Back injuries at any age are terrible. I have known a few nurses who damaged their backs on the job and, like you, at a young age. I am glad to hear that you maintain a positive attitude towards life and hope you can enjoy your dream for many years to come.

Before I suggest what to wear, might I first suggest you (re)think your chicken coop and run setup. I'm 61 years old now, and when I built my coop and run, I designed it so that I never have to go inside the coop or the run to feed, water, and gather all my eggs. I can do everything from outside in my regular street clothes and never get them soiled. I don't need any special clothes or shoes because they never come into contact with chicken stuff. Of course, I do wash my hands after refilling the feeder, waterer, and collecting eggs every day.
A couple times a year I do clean out my chicken coop and put down fresh deep litter. On those days, I will usually wear some old clothes and either an old pair of tennis shoes or my mud boots. Cleaning out the coop can be a dusty process, so I also wear a face mask. When I get back into the house, I strip off the clothes and throw them in the dirty laundry by the washing machine - so they don't really get into the house. My shoes or boots can be taken off either at the door or in the garage. No big deal. Because I can get pretty dusty after the spring cleaning, I jump into the shower to wash out my hair (what's left at my age) and wash down good before putting on clean clothes. Dear Wife does not want chicken dander on our furniture, but neither do I. Like I said, I only clean out the deep litter maybe twice a year.
Old tennis shoes or mud boots work for me. Don't know your setup, but make sure you have shoes or boots that don't slip where you need to work. Protect that back as best you can. After my uncle slipped on some ice and broke his ankle, he bought some slip on rubber booties with small spikes that fit over his shoes. Easy to put on and take off, but ensures no slipping on the ice. Something like that might work for you on wet slippery grass and the such. NO FALLS!