Juniper, like many needled evergreens, are incapable of producing new growth on older wood in the interior of the plant. (They don't have the dormant leaf buds the way boxwood, holly, and other broadleaf evergreens do.) Any older branches exposed by pruning will not fill back in, though any foliage on young branch tips near a gap may eventually help to cover it up. Therefore, if the stubs left by this pruning contain no foliage left, you can probably prune them all the way back to the trunk if you want them out of view faster than waiting to see if other growth hides that area over the long term. Junipers ideally should not be sheared, or if they are, need to be pruned very carefully (only sticking to the youngest tip growth) to avoid this kind of damage, so hopefully someone can teach the lawn service personnel not to make that mistake again. We have seen a mature 'Grey Owl' shrub maintained with light pruning to a size smaller than it would normally mature without trimming, but in that case the cuts never were made far into the branches where they were leafless; live foliage was always left where the cuts were made to fill back in the following spring.

As far as timing goes, you can do any other desired pruning now if you wish -- avoid autumn, though -- or wait until early spring next year.

Miri