Is there a rigorous definition for this concept? ‘The collection of every individual thing’? If an ‘individual thing’ is something that is different from something else, ‘everything’ could be the collection of every x that is not y where x≠y?
The set U of all things: For all x, x is an element of U.
Unfortunately, U can be shown not to exist. Apparently every set must exclude something. (See Russell's Paradox)
To ask, What does 'everything' consist of? is to ask in effect, What things exist? This is the fundamental question of ontology. There have been many attempts to make this question rigorous by tying ontology to logic. One of the most popular approaches is to say that quantified formulas of the form (∃x)Fx are true if and only if there exists an object in the domain of quantification which when substituted for x satisfies the open formula Fx. Quine summarizes this position with the aphorism: To be is to be the value of a variable. 'Everything' is then the entirety of our domain of quantification.
Of course, this leaves unsettled the question as to what should be in our domain of quantification. We could be minimalist and hold that it should contain only those things that are necessary for a scientific account of the universe, but this places a huge burden on the reductionist program. Who is to say what is absolutely necessary, and for what purpose?
Restricting our logic to first order is perhaps too limiting: we might think it reasonable to quantify over properties or classes or propositions. Do numbers exist? Mathematicians quantify over them, but they don't seem to be the same kind of thing as chairs and kangaroos. Do events exist? Davidson proposed that we quantify over events in order to explain how "John ran quickly" entails "John ran". Do fictional entities exist in some extended sense? Meinong thought so, and free logics permit an extended domain of quantification containing fictional entities. Do minds exist? Or are they reducible to or identical with physical things? Do the fundamental particles of physics really exist? Physicists quantify over them, but pragmatist philosophers hold that they are nothing more than useful fictions that serve to aid us in making predictions. Do possible worlds exist? Many logicians and philosophers of language quantify over possible worlds, but only a few such as David Lewis are willing to allow that they have real existence.
So, even when you think you have found a rigorous criterion, most of the interesting questions about ontology remain.