Johnson said the deputy chief of UK's defence staff at the time, Lieutenant General Doug Chalmers, had told him a raid using small boats to cross the Channel and navigate Dutch canals would be possible - but warned him of diplomatic repercussions.
According to Johnson, Chalmers - who has since retired from the military - told him it would not be possible to carry out the mission undetected and that "if we are detected we will have to explain why we are effectively invading a long-standing NATO ally".
"I secretly agreed with what they all thought but did not want to say aloud: that the whole thing was nuts," Johnson said in an extract from his memoirs that was published in Saturday's Daily Mail newspaper.
Neither the UK's defence ministry nor Chalmers - who currently serves as chair of the government's committee on standards in public life - had an immediate comment on Johnson's account.
The COVID-19 vaccines under dispute were developed by the University of Oxford and AstraZeneca but doses were manufactured by subcontractors in the Netherlands and the UK.
In March 2021, the vaccines were widely used in the UK but doses made at the Dutch factory were still pending EU approval.
Both the UK and the EU had contracts with AstraZeneca for vaccines and the EU sought to hold back finished vaccines at the Dutch plant for its own future use.
Johnson - who won election in December 2019 on a promise to conclude drawn-out talks to leave the EU - said he believed EU officials were acting under pressure from French President Emmanuel Macron.
"After two months of futile negotiation I had come to the conclusion that the EU was treating us with malice and with spite ... because we were vaccinating our population much faster than they were," said Johnson, who was forced out of office in 2022 following scandals including breaches of pandemic lockdown rules.
