Q.Holly bush limbs are black
When trimming bush, we noticed that the limbs are black and we have what looks like white eggs on the branches and under the leaves. What would cause this?

This may be Mealybug and a fungus both occurring on your plants.
Neem Oil is a good treatment as it works as both a fungicide and insecticide.
Here are a few links for you.
https://www.gardeningknowhow.com/plant-problems/pests/insects/mealybugs-control.htm
https://www.gardeningknowhow.com/ornamental/shrubs/holly/diseases-of-holly-bushes-pests-and-diseases-damaging-holly-bushes.htm
https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm