Tractor supply sells live catch traps that will work for rats if nothing else fits your need
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I'm a bit confused as to why you want to get rid of rats, but keep the squirrels. Squirrels will do all the same damage that a rat will, with the exception of tunneling. They steal feed, eggs and chicks just the same as rats do. Snap traps baited with peanut butter and cracked corn, then put where other animals can't get at it works for me.
We have other critters around also. We set traps after everyone is locked in coops, kennels, house, etc. Set them after dark and my DH gets up at 4:30 and checks them. Picks them up for the day. Rats are more active at night and that's when you will catch more of them anyway.
Never realized killing rats was so difficult. One shotgun, One shot, One of them gone. NEXT