Nothing Doesn't Exist
First, the definition that I’m using is quite literal, with nothing meaning “the lack or absence of anything.” It’s hard to define nothing in any non-negative way, but I think we all have a good enough idea of what nothing means to agree the above definition works pretty well. I’ve heard this called “ontological nothing”, and that’s how I will be using the word.
The interesting thing about nothing is that it does not exist. Don’t believe me? There is no point in the universe that is not something. “But what about empty space?” you ask. Space (or space-time, to be more precise) is most definitely a thing. It’s modified by the presence of matter, which is definitely something that nothing wouldn’t do. It’s also filled with “quantum fluctuation”, constantly generating so-called “virtual particles”, a particle and its anti-particle, which spontaneously generate and then annihilate. This only happens on the smallest of scales, but there’s still something there.
So what about outside the universe? There are two outlooks you can take there. The first and more traditional is that the question “what is outside the universe” is meaningless. This position holds that the universe we occupy is all that exists, and we can’t possibly leave it (even if it’s finite). The second is that we occupy a “pocket universe” surrounded by false vacuum, which is infinite. Again, asking what’s outside that is meaningless.
Just in case you don’t buy it, my justification for the assertion that “beyond the universe” is meaningless is to say that the universe is the total extent of space-time. Because it lacks space-time, the external universe would be dimensionless, and as such would be completely devoid of meaning in any way we can conceive. Also, because the universe is defined as everything that exists, asking what exists outside of it is pretty clearly meaningless, rendering the question unintelligible.
The conclusion is inescapable, nothing doesn’t exist.