The Simple Direct answer is YES wear eye protection, and use "Effective" dust collection !
All Walnuts contains Juglone. This chemical is a toxic defense mechanism that kills any competing tree in it's range. It is an extremely powerful herbicide, second only to Nicotine in effectiveness. Never dump Walnut shavings or sawdust anywhere you want plants to grow.
It is particularly lethal to Horses ! This is one of the reasons that Good Walnut can be so hard to come by as many mills will not mill walnut because they have to clean and sterilize the mill afterwards.
One of a mills greatest expenses and chores is waste disposal of the shavings. Clean shavings can be sold at profit to stables, and pet supplys. Tainted shavings are toxic Rubish !!! Consequently Burning Walnut will release it directly into the surrounding enviornment.
All people will reacty differently, but as you have already discovered the walnut dust will get in the eyes, and be carried to and through Mucous membranes into your body. Any moist orifice will Do !

It usually takes a week or two to get out of your system, but could easily perpetuate a sinus infection.
I'm Violently allergic to Pine Tar, and Turpentine. Pine dust mixed with sweat on a hot day will burn like acid, and Any pine cleaner can put me in the emergency room in minutes. This is why I work with hardwoods predominantly, but Walnut, Rosewood, Cocabola, Ebony, Lignum Vitae, Sapele, and Mahogany etc... can be even worse, if not fatal.
My down draft table runs a dedicated 1250 CFM, and all my sanders are hooked up to an 8.5 horse Shop Vac. I still wear dust masks and respirators. My spray booth runs an addition 1250 CFM dedicated turbine, and most carving tables have an additional shroud. I enjoy working with wood, and prefer to keep doing it as long as possible.
Many of the solvents we use for finish are derived from wood products and petrol. Over time these solvents can act as a vehicle on unprotected skin, and pull any substance into your system until you reach a a saturation point and then you become hyper-sensitive to it. This condition is known as Enviormental Disease. Simply Stated a person CAN Become allergic to many of the things we love the most. Hobbies, pets, foods etc... So All solvents should be handled with care, and Any Sawdust is Supect. Not to Preach, but certain woods are particularly Lethal ! Walnut is a lesser known risk that most folks are not aware of.