First,
Call your area's Conservation Warden. They will be glad to explain in detail what you ask.
I am not from CT so I searched your states regulations.

Season Dates

• Gray Squirrel: Jan. 1 – Feb. 29 and Sept. 1 – Dec. 31
• Cottontail Rabbit: Jan. 1 – Feb. 29 and Oct. 17 – Dec. 31
• Snowshoe Hare: Jan. 1–31 and Nov. 21 – Dec. 31
• European Hare: Jan. 1 – Feb. 29 and Oct. 17 – Dec. 31
• Woodchuck: March 16 – Nov. 16

Bag Limits

• Gray Squirrel: Daily (8); Season (40)
• Cottontail Rabbit: Daily (3); Season (25)
• Snowshoe Hare: Daily (2); Season (10)
• European Hare: Daily (1); Season (10)
• Woodchuck: No Limit

Gray Squirrel has specific season dates. That means during that time Gray Sqirrel can be harvested. You are allowed to shoot 8 per day until you reach 40 total. Eat one or 2 or give away 1 or 2 so they are not on your premises and you can shoot 1 or 2 more. But your possession limit is 40.
Woodchucks, shoot all you want No limit!

But do call your game warden
FIREARMS
Your CT regulations say nothing less than a .22 cal (the .17 hmr is considered a .22 in your state)
So any air rifle .22 or .17 can be used to hunt squirrel or woodchuck
You DO NEED a hunting license on your OWN property from what I could find. In some states this varies. In my home state squirrels and woodchucks are considered a nuisance and I dont need a license on my property.
But we do have a squirrel season. Quite frankly my Red Squirrels disappear year round.
JW