Mariano's comment has essentially answered the question, but I'll go ahead and flesh it out.
On any manifold $X$, there is an isomorphism of groups $\mathrm{Pic}(X) \xrightarrow{~\cong~} H^2(X; \mathbb{Z}),$ $L \mapsto c_1(L).$ Now if $c_1(L) \equiv 0 \pmod 2,$ then there is some element $a \in H^2(X; \mathbb{Z})$ such that $c_1(L) = 2a.$ The above isomorphism tells us that there exists a complex line bundle $K \in \mathrm{Pic}(X)$ such that $c_1(K) = a$ and $K \otimes K \mapsto 2a = c_1(L).$ Then $K \otimes K \cong L,$ so that $K$ is a square root of $L$.
For a proof of the above isomorphism, see for example Proposition 3.10 in Allen Hatcher's unfinished book Vector Bundles and $K$-Theory.