This must be due to radiation exposure which declassified French documents said was higher than known at the time, said Mr Mitchell.
The Herald asked Mr Woodhouse whether the Government had considered an inquiry into the health of the frigates' companies and the children of those men.
"While there has been no evidence produced that calls for such a study to be undertaken, the Government may be open to considering this course of action should some evidence be presented," Mr Woodhouse said.
He said 70 veterans who served on the frigates were receiving a War Disablement Pension.
"I would encourage anyone who served on these vessels and who is suffering from a medical condition they believe to be attributable or aggravated by their service to contact Veterans' Affairs New Zealand so they can receive the assistance they are entitled to."
The pension was a lifetime entitlement and recipients were entitled to free medical treatment in relation to each disability on a presumptive list. However, Mr Mitchell said it became apparent at the reunion that quite a few of the men did not know the pension was available for their illnesses.
He said the minister was "burying his head in the sand" by demanding veterans provide proof of a need for an inquiry.
The most important question for the men was whether they had passed on the effects of radiation to their children and grandchildren and what help would be given to them.
"We want to turn it around and say, 'we claim, believing that it is'. We even go to doctors to sign a medical certificate to say they believe, or know it is, related in order to receive the pension.
"It's time the Government believed us and proves that it is not radiation-related."
Mr Mitchell said veterans believed they were exposed to radiation in the sea water, which the ships desalinated and supplied for use on board. Veterans also believed radiation affected ships' food supplies, such as potatoes, because they were stored in outside lockers on the deck.
Returned and Services Association national president Don McIver said he understood that veterans were concerned how long a study would take when time was running out.
"The RSA has great sympathy for the nuclear veterans and I understand why they don't feel they have been properly treated at this stage."