As of what I experienced, no they don't change their happiness in different trees, it's purely staggered dominance, keeping your squirrels thirst and hunger up and the wolf's affinity that determines it ^^
.
when they become unhappy, your squirrels' health won't heal as fast when they get hurt in exploring~ though it's easy to heal them by popping a quick Health Potion, Team Heal, Robin Fstuff cheeksher or Blue Jay fstuff cheeksher :)
.
Personally I am a 0 happiness pack all the time and I never needed that option, if I really need to heal a load of squirrels there is a hot spring under Pack > Food and Drink, that instantly maxes all squirrels of Health, Hunger, and Thirst once a day