Weak Posterior Chain solution?

From everything I have read it sounds like you have major flexibility/mobility and stability issues.
Your hips and thorasic (upper back), and shoulders are not mobile enough; your lumbar region (ab region) is unstable.
Fix your flexibility and mobility issues, and stabilize the lumbar region and you will have a drastically improved squat.
Healed shoes can help with the flexibility issue but try to fix it too.