A group has odd number of elements iff each element is a square, $a=b^2$.

If I understand you right, then no. Just because $n$ is even does not mean there is an element $a$ where $n$ is the smallest power of $a$ that gives you $e$. For example, $G=C_2\times C_2$ has order $4$. And of course, $a^4=e$ for all $a\in G$. But also $a^2=e$ for all $a\in G$.

If $n$ is even, there is at least one non-identity element $a$ where $a^2=e$. So the map $G\to G$, where $x\mapsto x^2$ is not one-to-one. So the image of that map cannot be all of $G$.