Can a burning bush be propagated from cuttings?

You Americans and your 'Burning Bush' Which one?

If you mean Euonymus alatus then then you've just missed the optimum time of Sept.

Take 3 to 4 inch lateral cuttings, with a 'heel',and line out in an equal mix of peat /sand in a cold frame.

They should have rooted by next May, than line them out in a bed for 2 years. Job done.

Or; you can sow their seeds,but they can take a year and a half to germinate.

I'm really not sure but if you want to try you need to take a tip cutting of new growth in spring. Take a cutting of about three inches and take off the bottom two leaves. The roots will grow out of the area called the node where the leaf grows off of the stem.

Use a rooting hormone for rooting softwoods. Dip the end in water and then in the rooting powder or liquid and then plant in good potting soil in a three inch pot. This needs to be done in the spring, not now because there is not enough light for the cutting to grow this time of year.