Note: I have never heard this before, I'm only reasoning through it, so don't take my answer as the answer.
This problem is defined by its ambiguity. Can ANY number be in either box, from negative infinity to infinity, with equal distribution? It doesn't say anything about it. From 0 to infinity? Only integers?
If it's negative infinity to positive infinity (integers or not) you can do better than 50% only in a trivial sense; If you see -5.6 billion in the first one there are infinite numbers in either direction, so you have a 50% chance either way. So, no, you can't do better than 50% given the equal-distribution-negative-infinity-to-positive-infinity-assumption.
If there is a known discrete range, than it's obviously easy to get better than 50%. But with any infinite range, you can't do better than 50%.
Note: I am really guessing here that this problem was formed with this thought: If the range is 0 to infinity, then whatever finite number I get in the first box has a finite number of numbers smaller than it, and infinite larger; therefore, I should always pick larger and I am correct some number approaching (but not exactly equal to) 100%. This is an absurd conclusion since it implies that no matter which box I pick first, it will always have the smallest number, when the fact is this will only happen 50% of the time.
Not being a mathematician, my guess is that the unstated assumption that leads to this contradiction is what the comment below points out: The entire idea of a uniform distribution over an infinite set is impossible.
So what does this say about the question? Well, just that it doesn't really make sense with an infinite range, and is horribly simple with a finite range.