While this is not a government, ownership is by a government agency. What is an appropriate response in this case?
It depends on how your state defines local governments. In Texas, for example, it is by statute that hospital districts can be created that have taxing authority which makes them a local government.
I do not know the context of your question (and the answer may vary by state), but I am inclined to think (without any other additional information) that in Texas, at least, you would work for a local government.
The State of reference is Nebraska in this case. Follow up question, how does the government status question influence the tax form? That is, what changes does it cause marking government employee or non?
When you entered your W-2 from the hospital, in box 12 you must have entered a "G" which is elective deferrals to a 457(b) plan. 457(b) plans are offered by state and local governments and some nonprofits. Since you are working for a hospital, a non-profit, you would answer "No" to the question regarding working for a state or local government.
Saving for your retirement is a good idea with your contributions growing tax deferred. Plus some employers match the amount you contribute and depending on your income level, you may be eligible for a Retirement Savers Credit from the IRS.
Click HERE for more information on the Retirement Savers Tax Credit.