Miller probably charge their patient on Saturday but the actual "work" of charging is done on a non Sabbath day. So what we are talking about here is the actual work of charging patients. Well evidently Miller as what you say does not have any accounting and cashiers on Sabbath. So there is definately no actual "work" of charging done on the patient on saturday. Even the actual "work" of computing the charges are not done on Saturday. And certainly there are no payment or bills actually being done on saturday.
So to put it in a simple way there is no work done on Saturday on behalf of the Staffs and patients.
The purpose of the Sabbath:
The Sabbath and the family were alike instituted in Eden, and in God's purpose they are perpetually linked together. On this day more than on any other, it is possible for us to live the life of Eden. It was God's plan for the members of the family to be associated in work and study, in worship and recreation, the father as priest of his household, and both father and mother as teachers and companions of their children.
The Sabbath is the day above all others when we should acquaint ourselves with God through His works. In the minds of the children the very thought of the Sabbath should be bound up with the beauty of natural things. Happy is the family who can go to the place of worship on the Sabbath as Jesus and His disciples went to the synagogue--across the fields, along the shores of the lake, or through the groves. Happy the father and mother who can teach their children God's written word with illustrations from the open pages of the book of nature; who can gather under the green trees, in the fresh, pure air, to study the word and to sing the praise of the Father above. By such associations parents may bind their children to their hearts, and thus to God, by ties that can never be broken.
Sabbath is for the people not for businesses and Hospitals.
Please don't misunderstand these. So on the patients end and on the staffs end despite the charges on Sabbath they should still be able to fulfill the sabbath without burden. It will even give them more leeway in observing the Sabbath by ceasing work on Sabbath.
To be more realistic.
Hospitals are not Jesus
If you want to take advantage of free hospitals then it is non existent here in this Philippine economic setting. Even Vicente Sotto hospital is not free.
But If you have faith in Jesus he can heal any sickness for free.