Thanks for writing. If you have a large container (48 inches or more in diameter), your euonymus should be able to survive the winter outdoors. But if your container is smaller than that, it's best to move it to a sheltered spot, such as an unheated garage or shed for the winter.
It doesn't require any special care --- except you do need to water it periodically if you move it to a spot where it won't get snowed/rained on. Don't let it completely dry out over winter.
---Justin, Senior Garden Editor, BHG.com "