Winter die back is relatively common from time to time and can result in an entire branch failing, although it is much more common with tip die back. It could also be showing some signs of drought stress.....didn't you have a pretty dry summer last year? Regardless, just remove the dead wood cleanly. And if you have another hot, dry summer, a couple of good deep soakings would help.

If a branch does not leaf out by June I would say you are safe to prune it off. Clean your pruning tool before and after cutting to prevent spreading any possible disease. Just good practice.