Mr Cleverley told The Star the policy is "normal" and he stands by it. He said the board is similar to a board of directors and is therefore not required to report back to the community.
"You (board members) are responsible and accountable not to the people, they elect you, [but] you are responsible to the shareholder, the Ministry of Health," he said.
Of the 11 board members, seven are elected by postal ballot with four appointed by the Ministry of Health.
All seven current board members are seeking re-election in October. Twenty other candidates are contesting the election.
Ms Kane felt "gagged", but found ways around it by mentioning things at the meetings, hoping the media would pick up on them.
"It is archaic, it's crazy. The first thing you are told after the elections is you are not responsible to your community - you are responsible to the minister," she said.
But Mr Cleverley said the board was about "good governance, rather than what is a democratic society".
Local government political expert and Lincoln University lecturer Jean Drage said it was "utter nonsense" the CDHB was like a board of directors.
"(Elected members are) elected by the community so they take the views of the community with them when they go to board meetings and are accountable back to the community," she said.
She said district health boards need to be "more responsive" to communities that elected them.
Minister of Health Jonathan Coleman would not answer questions from The Star about the policy.
Ms Kane said she was not surprised as Dr Coleman had never met with board members.
"I don't have any respect for him. What is worse, he is being advised only by the ministry and not those in Canterbury," she said.
Dr Coleman said he had met with the CDHB's chief executive, clinicians and chairman. He would not say why he did not meet with the board.
CDHB elected member Anna Crighton said she was unaware of the policy, and said it needed to be reviewed.
She said Mr Cleverley's statement was "ridiculous" and he was answerable to the "people of Canterbury".
Elected board member Aaron Keown said he wanted the policy gone while candidates Kelly Dugan and Janet Du Lu said it should be reviewed.