Quine is simply presenting a definition of 'everything'. Think of it this way. If I ask 'what is there?', a correct but unhelpful answer is 'whatever there is'. You cannot disagree with that answer, since it is really just a re-expression of the content of the question. If you define 'everything' as meaning 'whatever there is', then 'everything' becomes the answer to the question 'What is there?' by definition, so you cannot disagree with it if you accept that definition of 'everything'. Of course, you can adopt the view that 'everything' has some other meaning, and therefore disagree with Quine on that basis, but then you would be making the mistake of arguing at cross purposes.

Likewise you can define 'everything' to mean 'all that exists', in which case you cannot argue against the statement that everything exists except by arguing about the definition.

It is a truism to say whatever exists exists, and it is hardly less of a truism to say everything exists if you define 'everything' to mean 'whatever exists'.

Quine is simply presenting a definition of 'everything'. Think of it this way. If I ask 'what is there?', a correct but unhelpful answer is 'whatever there is'. You cannot disagree with that answer, since it is really just a re-expression of the content of the question. If you define 'everything' as meaning 'whatever there is', then 'everything' becomes the answer to the question 'What is there?' by definition, so you cannot disagree with it if you accept that definition of 'everything'. Of course, you can adopt the view that 'everything' has some other meaning, and therefore disagree with Quine on that basis, but then you would be making the mistake of arguing at cross purposes.

Likewise you can define 'everything' to mean 'all that exists', in which case you cannot argue against the statement that everything exists except by arguing about the definition.

It is a truism to say whatever exists exists, and it is hardly less of a truism to say everything exists if you define 'everything' to mean 'whatever exists'.

Addendum

To cover your comments on my answer...

When Quine says 'everything' is the indisputable answer to the question 'what is there?', I say it is sensible to assume that by 'everything' he meant 'everything that there is', ie that he was employing a deliberate tautology. You seem to be suggesting that by 'everything' he instead meant 'everything that there is and some things that there are not'. Since that would render his claim patently wrong from the get-go, what on Earth makes you think he would be so stupid as to have intended your interpretation of his words?

Quine is simply presenting a definition of 'everything'. Think of it this way. If I ask 'what is there?', a correct but unhelpful answer is 'whatever there is'. You cannot disagree with that answer, since it is really just a re-expression of the content of the question. If you define 'everything' as meaning 'whatever there is', then 'everything' becomes the answer to the question 'What is there?' by definition, so you cannot disagree with it if you accept that definition of 'everything'. Of course, you can adopt the view that 'everything' has some other meaning, and therefore disagree with Quine on that basis, but then you would be making the mistake of arguing at cross purposes.

Quine is simply presenting a definition of 'everything'. Think of it this way. If I ask 'what is there?', a correct but unhelpful answer is 'whatever there is'. You cannot disagree with that answer, since it is really just a re-expression of the content of the question. If you define 'everything' as meaning 'whatever there is', then 'everything' becomes the answer to the question 'What is there?' by definition, so you cannot disagree with it if you accept that definition of 'everything'. Of course, you can adopt the view that 'everything' has some other meaning, and therefore disagree with Quine on that basis, but then you would be making the mistake of arguing at cross purposes.

Likewise you can define 'everything' to mean 'all that exists', in which case you cannot argue against the statement that everything exists except by arguing about the definition.

It is a truism to say whatever exists exists, and it is hardly less of a truism to say everything exists if you define 'everything' to mean 'whatever exists'.