No. Gurgi's species is never formally identified, but he is not a dog. An actual dog features as a minor antagonist, chasing the heroes during their initial escape from the Horned King; other than narrowly avoiding being crushed by a castle gate, it does not come to any harm.
Hen Wen the pig is subjected to various abuses when she is captured by the minions of the Horned King, including being chained up, having her face dunked into water and being threatened with a hot coal.