Most likely the roots have survived so cut off the damaged sections and protect it at night and you should see signs of new growth on that begonia in a week or two. Fertilize with a slow release plant food like Osmocote now to help get the new growth going.  Some annual or bedding plants are very cold sensitive so wait until all danger of chilly nights below 45 degrees has passed before setting out begonias, coleus, marigolds and other warmth loving plants. Keep growing, Marianne Binetti