If I may, I'd like to supplement the answers by @rerun and @Berin.
They both speak to the same concept, that running shoes might not provide you the most stable platform (since they can be squishy). I am not arguing that point, although I feel it's a generalization and should be treated as such - but it's a generalization that seems reasonable.
So, let's assume the shoes do make you more unstable.
This "instability" isn't necessarily a bad thing. All it means is that you will need to invoke some extra muscle power to counteract the effect. Your muscles are already working to "stabilize" you, but now they have to work a bit harder because you've chosen to increase the challenge to your stability. You could add even more instability by doing a squat on a say, a wobble board.
If you are aware of and prepare for that effect - perhaps by lowering the weight significantly until you are rock-solid in your lift - that instability may be something you want to add to your lift because it can give you a new challenge (resistance training is all about progression, right?).
It depends on what you are trying to achieve.
If your goal doesn't involve controlling the instability the shoes might be providing, then remove the instability. But if your goal is simply to lift as much weight as you can lift, without regard to the rest of your body, then the instability is an extra variable you don't need.
Just like guns don't kill people, shoes don't harm bodies by themselves, but how a person uses them in a given situation can certainly cause unexpected and unwanted results. This is true for ANY piece of gear or equipment.
You need to think about what you are trying to accomplish, and adapt your squat accordingly.
One more comment related to the shoes tilting you forward that @rerun mentioned...
The range of motion in a squat varies depending on the person doing the squat. My range of motion isn't necessarily the same as your range of motion. The bones in your body have a lot do to with your range of motion and at what point your hips start to rotate as you descend (which means you've exceeded your range of motion).
One way to counteract this - for people with a very short tibia compared to the length of their trunk and femur - is to put a small wedge under the heels. This has as similar effect that @rerun is describing, but it's really about increasing your range of motion during the squat artificially. Someone who isn't built this way physically might find that this causes more problems than it corrects.
I am certainly not recommending you do this, but I do point out that there are times when extra room under the heels can be beneficial.