Answer: Unfortunately I am not certain what you are describing. (Housefly type eggs are white, and blackflies lay eggs in water.) These shrubs are usually quite pest and disease free with the exception of occasional Japanese beetle problems. You might be seeing frass or excrement from a caterpillar or other insect. I would suggest you consult with your county extension to identify what you are seeing and based on knowing that, determine if you need to do anything. If it is something that requires a chemical control, they will have the most up to date recommendations on what to use and how/when is best to use it. I'm sorry I can't be more specific for you.
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