In discussing competitive markets, what rate of output is appropriate for a "nonprofit" corporation (e.g., a university or hospital)? 

Non Profit corporation as the name suggests does not work to get profits. They just focussed on expanding their yield . For the non profit corporation, take example of corporation like university and hospital. They both work for the welfare. Hospital work for benefits of patients and universities work for the benefits of students who are doing gradations.

These corporations have to make a strategy so that without getting any profit, they will overcome the expenses of services and infrastructure .

So, Their output level will be appropriate when their cost will be equal to market cost at zero profit.