The mention of "evidence" in addition addition to "proof" is a good way to start.
One can explain that the fact that the square of 2 is also even is evidence for the assertion/conjecture/hypothesis/claim made. The fact that the square of 6 is also even is further evidence for it.
And so on.
But no matter how much evidence of this form one generates there might always rest some possibility, some doubt, that the general claim could be false.
Then, one could recall that in court one common standard is to say that for something to be considered proved there needs to be evidence beyond a reasonable doubt; but for other situations one might only need to show a "Preponderance of the evidence" or still something else. See the Wikipedia site Legal burden of proof for various concepts.
Finally, one could say in mathematics the standards of proof are very strict and there needs to be evidence beyond any doubt whatsoever for something to be considered as proved.
To sum it up: also in everyday usage there is (or at least there should be) a distinction between "evidence" and "proof," and this is a good opportunity to recall this. Which quantity of supporting evidence is considered as a proof differs depending on context, and in mathematics is especially high.