Hi Yvette - These are spider webs in your boxwood. Spiders do not feed on or cause damage to boxwoods, but the webs can look unsightly as plant material accumulates in them. Spiders are generally very good predators to have around a garden because they eat a lot of pest insects. You can bush out the webbed areas using a broom or wear garden gloves and brush out the webbing by hand.
The brown branch on the right side of your first photo may have died from physical injury or a fungal disease (possibly Volutella blight) but we cannot say for sure. In any case, it is a good practice to prune and thin out boxwoods. This will stimulate healthy new growth and allow for more air circulation around the plant (to help reduce the chances of fungal disease issues). Some common insect pests such as boxwood psyllid also can be managed by pruning. Please watch our brief video for a demonstration of good practices for pruning and thinning boxwoods. https://youtu.be/3i8mcoKapeY
Also, visit our website for more information on boxwood care and how to prevent/manage common problems, https://extension.umd.edu/hgic/topics/boxwood-preventing-and-managing-common-pests-and-diseases