I know that red squirrels are protected I was wondering why? I've noticed many other states allow hunting of them. I am not aware of threats to them other than invasive squirrel species. As to the putting down of injured or sick animals I have put down animals before in my home state of Missouri mainly raccoons that showed symptoms of rabies.
Certain species of ground squirrels can be hunted in Idaho. The most commonly hunted varmints in Idaho are the Columbian Ground Squirrel and the Yellow-bellied Marmot "Rockchuck". There is a page in our Upland Game Rules that talks about squirrels and contains a link to a field guide.
Do not put any animal down without notifying appropriate authorities first.
Here is a list of those which cannot be hunted at this time.
Chipmunks -- Neotamias spp
Columbia Plateau (Merriam's) ground squirrel -- Spermophilus canus vigilis
Golden-mantled ground squirrel -- Spermophilus lateralis
Great Basin (piute) ground squirrel -- Spermophilus mollis artemesiae
Northern flying squirrel -- Glaucomys sabrinus
Red squirrel -- Tamiasciurus hudsonicus
Rock squirrel -- Spermophilus variegatus
Southern Idaho ground squirrel -- Spermophilus brunneus endemicus
Wyoming ground squirrel -- Spermophilus elegans nevadensis