If an irreducible complex representation of $G$ is of quarternionic type, then it must have even order, and so the group order must be even.
However, is it further true that $G$ has order divisible by 4? So $|G|$ is never equal to $2$ mod $4$?
I have checked and this is true for $|G| \leq 200$.
This has been answered affirmatively on MathOverflow here.