Unlike mites, lice are visible to the naked eye. If you part his hair you should be able to actually see them.
The hair loss around his eyes looks like zinc deficiency to me. Some goats seem to need more zinc than others - I have a wonky doe with this problem who tends toward the balding eyes/nose/topline every winter without additional supplements. I used to give her human zinc tablets, but then I learned Zinpro makes a product called z/m that is just zinc and manganese with no additional copper. One teaspoon per day fixed her. I topdress it on her grain, but you can probably get the buck to eat it out of your hand if you don't feed him grain because it smells malty. It'll turn him around in a couple weeks if this is his problem, and if zinc is not his root issue, it still won't hurt him to try it.