Why are my dog's black nails turning white, brittle, and splitting?

Poor Ruger - I'm so sorry for his past difficulties as well as for what he's going through now - he's so lucky to have found you. The first thing I would want to rule out is ringworm in this location. I know it sounds strange, however it can get into the nailbed and cause the kind of brittle, discolored nails you're seeing. Ringworm is actually a fungus and not a worm as some people think. Your vet can run a fungal culture using shavings from the nails to check for this.

If the culture comes back negative, I would probably next consider whether this could be a case of lupoid onychodystrophy. German Shepherds are predisposed to this disease, and since Ruger is a mix it's worth considering. Treatment can be challenging, but it's worth trying several things including omega-3 fatty acids, antibiotics such as doxycycline, and pentoxyfillne to increase blood flow to the region. Some dogs will also respond to steroid therapy. Best of luck getting this figured out.