How do you explain paradoxes to non-mathematicians? For example, how do you explain why the perimeter of this staircase does not converge to $\sqrt2$? Or, why isn't $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$?
I would say, the reason is simply because they cannot be proved. But non-mathematicians don't find such explanation satisfactory. They seem to be more satisfied by sophisticated explanation. For the staircase paradox, my friend who studies physics reasoned that it's because the perimeter is of dimension 1 not 2. It makes no sense to me, but they found such explanation more sensible.

In the case of $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$ at least, there is no paradox if you understand the limits of your rules.
You and the non-mathematician learned at some point that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ for nonnegative numbers a and b. If one has forgotten this last part, then they might be amazed by $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$, because they have blindly applied a rule where it is not applicable.
The same thing might happen if you pointed out that $ab \geq 0$ for all natural numbers. If someone forgets the last part and is amazed by the apparent paradox $(-1)5 \geq 0$ , does this really deserve to be called a paradox? 
Paradoxes do not have anything to do with "not being able to be proven", they are roughly just instances of where something you would expect to be true is not, in the system you are working in.
Like the "twin paradox" in physics: if you are (erroneously) under the impression that time is absolute, you are stumped, but if you believe in relativity then everything is OK.
