"The Boards have an obligation to not keep their meetings secret. I believe that is a requirement by the IRS. If there was ever an audit we could face some serious penalties."

That is not correct. The requirement only applies to Government run 501c3's and those incorporated in Florida. It's clear in the code.
'Most' non profit meetings are secret, that is the norm unless their constitution states otherwise. Doesn't take much googling to prove that.