When should I add meat/vegetables when cooking congee?
The process of cooking congee is similar to the process of making broth, but with rice added. I am not sure why one would want to use boneless chicken breasts in congee because this will impart minimal flavor (I would guess this is a Westerner trying to reduce fat). I am more familiar with using poultry or fish carcasses and aromatic vegetables to make a broth, and then straining and using this broth to cook the rice. For a faster way, you can add rice into the stock while the broth is cooking, but then you have the bones to pick through. 

Traditionally, congee is often used as a dish for leftovers. You throw things in that you have on hand and often already cooked from the night before. Many of these are simply added when served. For example: leftover stir fried meat, cooked vegetables, hard boiled eggs, preserved eggs, fresh green onion, fresh cilantro, or fried dough. 

To summarize, if you're using the ingredients to flavour the broth, then add at the beginning. If not, then add when serving.