 Odd unit group forces characteristic two

In a nonzero unital ring with finite odd unit group, the unit $-1$ has order dividing two. By Lagrange's theorem, that order also divides $|U(R)|$, so $-1=1$ and the characteristic of a ring is two. Consequently $\overline{a}\mapsto a{1}_R$ defines a unital ring homomorphism ${\mathbb{F}}_2\to R$. This map still exists for the zero ring, whose characteristic is one under the least-positive-integer convention.