I have a loud squeaking noise and it happens at very low speeds while turning and or at below 10 mph. 2002 cavalier base coupe 2.2l non eco tech, AC is not on when this occurs.

 

Do you have hydraulic power steering? (pretty sure you do) Might be the pump or the steering box need repair or replacement.

 

To me this sounds like a bad serpentine belt or tensioner, I had similar problems to the ones you're describing and I ended taking apart my wheel wells to check the serpentine belt on my Cavalier. Lo and behold the belt tensioner was cracked in 5 places, got new belt tensioner and no more squeals when turning, If I were you I'd start there as it's probably the cheapest, and good luck getting anything in there it is the tightest space I've ever worked in. Here's a video of a step by step.