Michael Johnston survived calls for a vote of no confidence after the director and company secretary endured an uncomfortable annual meeting.
On top of that request by one shareholder, another urged Johnston to resign, suggesting he was a barrier to improving the club's position.
The requests were noted but deemed not permissible at the AGM.
Shareholders were told they would have to call an extraordinary general meeting to have them considered.
Kilmarnock lie second bottom of the Scottish Premiership and are favourites to face a play-off to avoid relegation.
They have also announced a £725,000 loss for the financial year to 30 May 2015.
The meeting marked the end of Jim Mann's reign as chairman, the businessman having succeeded Johnston in the role.
Mann, who had previously cited health, travel and business reasons for his decision, had not been expecting his last duty to be presiding over such a hostile affair.
"In some ways, I'm pleased I won't be repeating what was a bumpy AGM in a number of different ways," he told BBC Scotland.
"But there were a number of issues that were raised from the floor that the club's going to have to take away and deal with."
When asked if he expected the disgruntled shareholders to pursue the matter, Mann added: "Who can tell? That's down to someone to raise a grievance and send in a document asking for an EGM to discuss it.
"So I'll leave that to the people who were here and whoever is on the board in the future to deal with."
The meeting, which lasted more than two hours in the Park Hotel, was attended by 500 shareholders.
Manager Lee Clark was well received when he handled a question and answer session about the footballing side of the club.
His message to his audience as he left the meeting early to take charge of training was: "Don't give up - we'll keep fighting!"
However, a presentation of the accounts by the club's new head of finance, Jim Millar, who was not in position during the previous financial year, did not go down well.
That prompted Johnston to step in to try to clarify matters, but that led to shareholders questioning his position at the club.
John Gall, managing director of Brownings Bakery, which is locked in a dispute with the club over the naming rights to "The Killie Pie", also took the opportunity, as an attending shareholder, to lambast Johnston's tenure from the floor of the meeting.
Director Billy Bowie, whose funding has been a major support to the club, said he was not going to ignore the shareholders' concerns.
He was applauded when he confirmed the board will hold a revue and get back to them.