Why did my chicken fingers not brown?
Try either a higher oven temperature or more time. Probably higher temperature, unless your chicken was undercooked. Browning occurs above 212F (the boiling point of water), so it won't start until the moisture (eggs and milk) has cooked off of the outside of the chicken.

You may also want to let the milk/egg mixture drip off of the chicken a little (so it isn't too thick) before rolling in the breading.