The issue was brought up by some supporter during the Q&A portion of the meeting.
Referencing sectarianism aimed at Hearts boss Craig Levein and Rangers manager Steven Gerrard, along with pro-IRA chanting, the shareholder expressed his concern that these songs have become more prevalent in recent years.
Speaking directly to the board, he said: “I feel there has been an increase in sectarian chanting including calling Levein an orange b*****d.
“‘Cheer up Stevie G...’ we are better than that. It shames us, it embarrasses us.
“The song originated from the Rangers support as sectarian abuse to Tommy Burns and now we are doing it. Embarrassing.
“I’m not putting all the blame on the Green Brigade but drums and megaphones bring the beat for these unsavoury songs.
“There’s a higher prevalence of these songs than five years ago. I don’t know what answer is but want board to act on it.”
Chairman Ian Bankier replied: “It’s very damaging and that sort of singing takes club to the gutter.”
Chief executive Peter Lawwell then added: “It’s very unfortunate that a small minority is dragging us down. It’s wrong for the club and gives enemies the chance to categorise us like everyone else.”