No, it's not possible to forbid a certain type of soldier from a specific mission.
It is possible to make it a bad idea to bring a certain soldier type on a mission using OpenXcom Extended+ - in the X-Com Files mod, trying to bring a dog on an underwater missions causes the dog to drown (and all your other soldiers to hate you). But that is a pretty involved bit of modding using the extra features of OXCE+, namely causing different missions to have sets of conditions that transform the armors of certain soldiers, and then giving an armor set negative HP regen.