what do I do?
I bought a Euonymus Alatus Compactus - Dwarf winged burning bush. I planted it in the fall. The leaves turned red and remained on until late in the fall. I put a protective screen around it for the winter. It didn't grow leaves this year and looks dry. There are small stems with leaves coming up from the roots..Should I prune back the bush and let the new growth stay or do I leave the bush as it is?
The long harsh winter that nearly the entire United States suffered likely killed off the shrub.
You can go ahead and trim off the dead material. Allow the plant to grow up from the root as it is doing. Do not trim or prune any of the new growth until the plant goes dormant this fall.
Here are a few links for you.