If you don't have auto insurance and you get into an accident can you sue the driver who hit you?

If you were legally required to have insurance, your position might be compromised a bit, but if the other driver was At Fault, and will not cover your costs, yes you could sue. Juries deciding liability usually will not be allowed to know anything about insurance coverage, because it doesn't bear on whether someone has a liability, but could sway them emotionally with the perception of a big pool of faceless money. The job of the insurance company is to cover the liability once it has been legally established.