It depends on who's calling names and what the intent is.
My mother would not allow nicknames. As a result, we were totally unprepared for the way kids at school nickname friends. Because it was banned at home, we saw it as behaviour intended to cause distress, and did not take being nicknamed in the spirit of friendship it was generally intended. Later on e did have some specific in-family nicknames, usually diminutives that were acceptable to the individual being named. But occasionally a 'pet name' applied by one particular person to one other would be used, and nobody else was allowed to use that name. My dad used to call me and my older sister "joe", for example. Never both of us at the same time, never in the same room. And nobody else could call us "joe".
But nicknames have to be acceptable to the individual being named. Some amount of thick skin is needed, but only some. Everyone has a right to be treated with respect.
If children are calling each other names, that has to be stopped. I step in and referee, point out that name-callnig is not driving the argument forward in any way. In fact, it risks stalling the argument at a "tis," "tisn't" level. Associated with this, is using "You" statements instead of "I" statements. It is more mature to say, "I do not like it when you say that," than to get angry and say, "You are mean to say that." The first leaves more 'wiggle room' for someone to put the brakes on themselves and take a more mature path. The second locks someone in to having to deny an accusation and get defensive, which only escalates things.
When I get in to referee, I hold up my hand for a stop, then say, "what is this all about, really? What are the proper issues?" Name-calling gets the argument off topic and this is why it stalls everything. Learning that this happens is a valuable lesson and takes repetition to learn.
If the name-caller is your husband, you are more hampered because he will not comply and is probably the main source of the problem - he sets the wrong example and then uses it as an excuse to get angry. And even though the kids hate what he does, they will still do as he does because they see it working - for him.
You have a lot of work to undo, methinks.
Marg