If you're still paying child support it appears unlikely that you "gave up (your) parental rights to (your) 14 year old daughter" as you contend, a process which normally would require a formal termination of parental rights by a court of proper jurisdiction.
And, therefore, if your child support order is still valid and binding, and it has now been properly amended to include a requirement that you place the subject child on your healthcare plan, you must abide by this order or appeal it, if there is still time to do so.
In answer to your Additional Information:
Again, I have to emphasize that I'm not familiar with Oklahoma or Nebraska law, since I'm licensed only in Georgia (born and grew up in Atlanta). Are you represented by an attorney who can address your situation? It would be best for you to find one if you don't already have one.
As both Mr. Hendrickson and I said previously, it sounds like all that really changed was >custodyshouldparental
You may be able top give up paternal rights in your state but you can never give up your paternal obligations which include child support up through the age of 18 in most states.
Even though your parental rights were surrendered, unles there was a specific direction that you would no longer be responsible for child support your obligation to pay child support and provide financially for your child continues until the time that she is adopted and someone else assumes the full legal responsiblity for her. If she is never adopted then you bear the financial responsibilty for her until she reaches the age of majority (in Nebraska this is age 19) or is otherwise emancipated. The father should also be paying child support.
Did you go to court for the court to direct health insurance coverage? If you did, did you present financial information about the cost. If you did give the court this information then you are likely to have little remedy. Parents are responsible to provide health insurance coverage if it is available and affordable. You can always file a petition asking the court to modify the health insurance order. You will need to show that you cannot afford to pay for the coverage. You need to go to court with information from your employer that shows what the cost of coverage is to you now and what the increased cost is to put her onto the plan. You may be able to get the Judge to determine that the cost is to great and that you are entitled to an order that says while it is available it is not affordable. You should take this step asap because you are unlikely to get money back after it is paid out for the insurance and the orders which direct the provider to set up the insurance are put in place very quickly.