Algorithm for computing square roots in $GF(2^n)$ Short question: is there an algorithm for efficiently computing square roots in $\mathbb{F}_{2^n}$?
Yes, there is an algorithm for efficiently computing square roots in $GF(2^n)$.

I don't know if this is the most efficient known, but the existence of an efficient algorithm can be shown by observing that squaring within $GF(2^n)$ is a bitwise linear operation, hence it is equivalent to taking the bit representation of the value, and multiplying it by an $n\times n$ matrix in $GF(2)$ to obtain the bit representation of the sequence.  Hence, it can be inverted by computing the inverse matrix, and multiplying by that.

Also, you have tagged the question with the "quadratic-residuosity" tag; however the field $GF(2^n)$ has the special property that all elements are quadratic residues; that is, for any element $x$, there will exist a unique element $y$ with $y^2 = x$.  Hence, in this field, the quadratic residuosity question is uninteresting.