Johnson has called for the EU to remove the Irish border backstop plan before the Brexit deadline of October 31.
Many of those who voted against former Prime Minister Theresa May's Brexit deal had concerns over the backstop - designed to guarantee there will not be a hard Irish border after Brexit - which, if implemented, would see Northern Ireland staying aligned to some rules of the EU single market.
Hammond, in his first comments since stepping down in July, said the early signs that Johnson could take the UK out of the EU with a deal in place were "not encouraging", the BBC reported.
"The pivot from demanding changes to the backstop to demanding its total removal is a pivot from a tough negotiating stance to a wrecking one. The unelected people who pull the strings of this government know that this is a demand the EU cannot, and will not, accede to," he wrote in the article.
He said Parliament would "make its voice heard", adding that a no deal "must not happen".
The BBC, citing a No 10 source, said the UK would leave on October 31 despite Hammond's "best efforts to the contrary".
The source added that Hammond, as Chancellor, "did everything he could" to block preparations for leaving and had "undermined negotiations".
The former Chancellor rejected this suggestion on Twitter, saying he wanted to deliver Brexit "and voted to do so three times".
Hammond also warned that a no-deal Brexit would "break up the UK", saying the reality would be "a diminished and inward-looking little England".
It was a "travesty of the truth", he said, to pretend that Leave voters backed a no-deal Brexit in the 2016 referendum.
"Most people in this country want to see us leave in a smooth and orderly fashion that will not disrupt lives, cost jobs or diminish living standards, whether they voted Leave or Remain in 2016," he said.
"Parliament faithfully reflects the view of that majority and it will make its voice heard."
Hammond's comments come as Downing Street said it expected a group of MPs to try to block a no-deal Brexit by attempting to pass legislation when Parliament returns in September, according to the BBC.