The Garlic Mustard Aphid does not use burning bush as a host. Unfortunately your photo is not high enough resolution for me to identify what is present on the leaves. There are other aphids that will feed on Euonymus. If they are aphids, then insecticidal soap is a good option. Sometimes you can even just spray them off with a hard spray from the garden hose. If you want them identified, you could send a sample to the MSU Plant and Pest Diagnostic Laboratory (pestid.msu.edu). 

The lady beetle larva is definitely a friend! They will eat the aphids. As bad as this looks, it will not have a big impact on the overall health of the shrubs. In the long term, natural enemies like the lady beetles will attack the aphids and keep them from returning next year. I would not recommend using any stronger pesticides. There is a chance some of the shoot tips that are most heavily attacked will wilt and die, but they can just be pruned off to improve the aesthetics of the shrub. The best course of action with this kind of infestation is usually to just ride it out and let nature take its course.