And Boycott believes captain Ben Stokes should also face the axe if he fails to learn some tough lessons with England staring down the barrel of a series whitewash.
Boycott, a long-time critic of McCullum's controversial 'Bazball' method, wrote in his Daily Telegraph column: "Hubris has taken over from common sense and that cannot be allowed to carry on.
"Stokes and McCullum are like men digging a hole to nowhere. If what you are doing isn't working then stop digging. Change is absolutely necessary to step up to the next level."
Addressing England managing director Rob Key, Boycott added: "What would I do? Change the coach. We are tired of this duo talking a good game but not delivering against the best teams so Rob Key, it is time for you to assert yourself.
"I would also sit down with Stokes and find out if he will compromise on his approach to batting and be prepared to embrace new ideas. If he doesn't see that a change of attitude is needed then you have to find a new captain."
Speaking on the Sky Sports Cricket Podcast, former England captain Michael Atherton described their capitulation as one of the most "disappointing" he could remember.
Atherton played in 33 Ashes Tests and never got his hands on the famous trophy. He continues to cover Tours as a broadcaster and journalist.
"I've played in a few bad ones, watched a few bad one, and seen a couple of whitewashes," he said.
"Actually, this is as disappointing as any for me, because I felt England had the tools to do the job here, or least challenge, and I felt there were weaknesses or areas to exploit in the Australian team."
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