State laws tend to differ but in general if injuries associated with an auto accident become apparent days after the accident you can still make a claim for them. You will typically need to get proof that the neck or back pain was caused by the auto accident in which you were involved in. This is done with a medical diagnosis from a doctor usually.
Under whose insurance policy the injuries will be claimed will depend on your state's insurance laws. In no-fault states typically these injuries would be report and claimed through your own insurance policy, even though you were not at fault in this accident since you were rear-ended. For instance, in Michigan your insurance company will normally pay for all "reasonable and necessary" medical expenses and the claim must be made within one year from the date of the crash.
If you are feeling pain contact your insurance company or the other party's insurance company so they are aware of your bodily injuries and also seek medical attention. Soft Tissue Injuries From A Low Impact Auto Accident is an article that might be helpful reading.