Pruning can stimulate new growth when plants are healthy enough to produce a second flush of foliage. Pruning the tip few inches of boxwood stems can also remove the majority of leafminers present. However, since insecticide for treating them was already applied, it might be best to just wait and let the plants leaf-out again when they're ready; it's not uncommon for boxwoods (if not pruned often) to just leaf-out once in spring, putting on a few inches or less of growth, and then being done with active growth for the year. (This pattern is typical of many shrubs, so not unique to boxwood.) Keep in mind that each trimming could potentially increase the density of foliage as regrowth occurs, which is not necessarily ideal for the health of the plant, since denser foliage could make the plant more vulnerable to a pest or disease outbreak. The advantage of more new growth would be mainly aesthetic for now, which is that it will be undamaged by mites or leafminer, so will look its best and help cover-up any remaining mite damage more quickly.

There probably isn't one right answer in this situation, but we'd lean towards not pruning and just letting them plants process the insecticide treatment and recover, since they will still be growing and developing leaf buds for next year's growth that will lie dormant on the stems until next spring, so at least they are still growing, albeit in a not very noticeable way. You don't necessarily need to fertilize, though a modest dose of a general-purpose formulation could be applied once you feel that the mites and insects are under control, since otherwise supplementing nutrients with an ailing plant can actually benefit the pests more than the plants.

Miri

You're welcome.