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The demand came at a protest march in Oamaru on Saturday against aspects of the proposal, Oamaru Hospital's current financial position and concerns over staff working conditions and patient safety.
Last Monday, Waitaki District Health Services (WDHS) announced decisions made after consultation as part of its proposed staff restructuring and layout change at the hospital.
About 150 people took part in the rally from the Oamaru Harbour area to Takaro Park, many with placards demanding WDHS chief executive Ruth Kibble and board chairman Chris Swann resign.
Speaking to the Otago Daily Times after the march, Waitaki Community Hospital Action Group chairwoman Dr Janice Clayton said the group, hospital staff and the community had lost confidence in Ms Kibble and Mr Swann, and change was needed if the hospital was to prosper in the future.
"We think they need to step down. I know they are doing what they think is right. They've done their best but when you start to lose the confidence from the group you are working with and you don't have their trust, you need to step down."
Former Waitaki mayor Alex Familton spoke at the protest and said funding for the hospital had been an ongoing issue.
He hoped the decisions made by WDHS went some way to "solving the problems" the hospital faced.
"There are problems. The board itself has been put under a lot of pressure. You have to realise in the last decade the facilitation of our services and the funding of those services have not been up to scratch for the last decade. You cannot put 10 to 15 years of neglect right overnight."
He was also concerned by specific areas of the document, namely some staff being asked to re-apply for their roles, which he said sent an "ugly message".
Group member Marcelane Ballantyne reiterated the group was "not opposed" to changes laid out in decisions on the proposal, rather the way it was handled and communicated.
"There is a few things we need from the management and board before we go ahead with these changes."
Those included enough skilled staff, less reliance on locum doctors and strong clinical training leaders.
She said the hospital should be marketing itself as the "epitome of rural healthcare".
"We are in a unique position to stand up and demand better."