Augie75 - I'm a victim of IL too, so I feel your pain.
You can kill squirrels just fine with .177, it's all about shot placement. If you want to go with a springer, a Diana 48 .177 will shoot about 20fpe. But a springer at that power is kind of a handful to shoot, and a bit heavy in the field. Even though it shoots with less muzzle energy, a Diana 34 which shoots closer to 15fpe is an excellent choice.
If you decide you just "must" have that 20fpe power, I would recommend a PCP. It will be lighter and easier to shoot accurately than a springer. If you can find a .177 Benjamin Discovery or Maximus, they are easy to fill with a hand pump because a low fill pressure is required. (I PCP pump looks just like a bicycle pump, but it can produce higher air pressure.)
I would not recommend a CO2 gun for squirrel hunting in Illinois. CO2 pressure is highly temperature dependent, so your pellets will change speeds as the gun warms up. (But they are awesome for indoor practice!!)
You didn't really elaborate on why you don't have an FOID (and you shouldn't feel any need to explain to us). Let me just say that if it is because of any past legal infractions, you should really check to see if you can have an airgun at all, or whether having a "weapon" would be held against you. I know it's pitiful to have to think like that, but it is what it is...