Answer
Let $a$ and $b$ be two even integers. Then by the definition of even, $a=2m$ and $b=2n$ for some integers $m$ and $n$. Therefore, $ab=(2m)(2n)=4mn$. But $4mn$ is divisible by $4$, since $mn$ is a product of integers and therefore an integer. Therefore, $ab$ is divisible by $4$ by the definition of divisibility. Since $a$ and $b$ were arbitrarily chosen, it must be that the product of any two even integers is divisible by $4$.
Work Step by Step
Note that "reusing" variables (e.g., letting $a=2n$ and $b=2n$) is prohibited, because then the proof would only be valid when $a=b$. For more on this, see section 4.1.