"I don't want to pay for someone else's health care" and "health care isn't a right" are two falsehoods that need to be addressed.
Firstly, the unavoidable reality is Americans have been paying for others' health care long before Obamacare.
One of largest factors behind rising health care costs comes from cost-shifting, which, simply put, is the percentage of the health care bill that comes from uninsured, under-insured, or government insured patients who do not fully cover the cost of their own service. For example, the average cost of an MRI procedure is around $2,600. This includes additional cost from those who flatly do not pay their bills, or those who have Medicare or Medicaid which does not pay the provider's cost of service some instances. This uncovered cost gets compounded year after year, resulting in the 10-15 percent annual increases in health care we see now.
While health care isn't a right under the Constitution, it has been a right under EMTALA since 1986. As those without coverage often cannot see a family doctor for the basic health management we all need, for things like illnesses and minor ailments, they seek it in the emergency departments where they cannot be turned away. Unfortunately the emergency departments are the most expensive place to seek care.
Repealing the insurance options offered in the ACA without an adequate replacement will further continue the dependence on emergency room care for many. The new Republican framework will, at present, do even less than Obamacare to address this issue and will ultimately result in universally increased cost and reduced access.