Wow, this is a very thought provoking question. My initial, gut reaction, is that healthcare is already being rationed - by health insurers, physicians, and hospitals/ERs. What tests and what medications used are determined not just by medical necessity but by cost/availability/acuity level of health condition(s). among others.
If you talk simply of rationing, there are various categories to discuss, including: age and when certain tests are no longer indicated; health conditions (liver or kidney failure and transplant criteria, causes of these failures, etc.; medication availability for specific conditions and who gets which meds; amount and availability of nutrition services and other ancillary health care services that expedite recovery; and more.
Based on the above, we already ration health care. More to your question, should we?
My simple answer, although I do not want to be too dogmatic, is YES. Cost overruns have afflicted the US health care system for decades and without some form of rationing, our population will pay over 20% and higher of the US annual budget on health care. Steps to keep costs down require collective effort, and less myopic oversight by health care entities. Topics for further discussion...

Hospital Price Transparency

CMS Hospital Price transparency is the healthcare industry equivalent of the automobile industry MSRP without informing patients about insurance rebate on price, cash down and quality of the 4 manufactured clinical outcomes.

VERY interesting and timely question. 
     Who is responsible:  it's always government.  Insurers or providers may have internal guidelines, but regulation and financial constraints instituted by the government rule everywhere health care is "public."
     How does it function:  Explicit is the only type I'm aware of on a large scale and it doesn't work.  The public simply doesn't accept explicit rationing even when they vote it in.  Implicit may work more effectively, but there are serious ethical questions, especially in America. 
     What are the relevant selection criteria:  QUALY is the overwhelming winner.  It's imperfect, but it's no less perfect than measures attempted by various states or nations and it's almost universally understood.  Attempting another measure could easily be considered a waste of time and money. 
     To what extent is current rationing just:  It isn't in my opinion.  Politics and the disease of the moment receive resources.  I would like to believe that a considered public health program would solve this, but I'm not aware of any instances where public health care isn't highly political.  Age is a very significant consideration.  Politics and culture easily reduce the focus on elder needs in many if not most instances. 
     What can be done to make it more just:  Equal advocacy.  That's not likely to occur, but equal spotlighting of all needs could make us a great deal more just. 
     How will health care rationing affect equal access to health care:  Evidently it will diminish equal access and increase wait times, etc.  That's certainly not the goal or expectation, but seems to be the result in current public health care systems. 
     What is the relationship in rationing and differences in health status:  Not good.  the concept is that resources are used for the many, hence funneled into preventive services and the "few" or very sick receive considerably less than they currently are in the US. 

The question isn't "should health care be rationed in the US," but "how do we control rationing and  can a private system function along side a public system."  Another great question is;  "How do we educated the American electorate so they can vote in a system they understand and that has a chance of succeeding."   

Thanks for the great discussion Dr. David E.M.