With the impact being very minor (Just over $500 in dmgs) I can understand how they do not feel that the accident caused your neck bone to be "moved outta its normal place" as you put it.

If this was the case the ER would have noticed this when they exmained you and obtained xrays.

If you have health insurance, or medical payments on your own auto policy, make the claim with your insurance carrier and let them attempt to collect the money back.

If you retain a lawyer to handle the claim for you, that still does not mean Geico will pay you.