It is not quite true. There is an algebraic gadget one can produce from a Spin structure on a surface with boundary (where we *fix* the Spin structure along the boundary), and one gets a bijection from Spin structures to such gadgets, but the process is horribly non-canonical. It does however let you prove that there are precisely two isomorphism classes of Spin surfaces of each topological type (for genus $\geq 2$).

This is discussed in Section 2.3 of my paper "Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces" (http://arxiv.org/abs/1001.5366), albeit it more generality (so-called $r$-Spin structures).

Briefly, the point is this: one ought to have some sort of algebraic structure on $H_1(S, \partial S ;\mathbb{Z}/2)$, and Johnson's construction works fine for elements represented by unions of simple closed curves, but for arcs between boundary components there is all sorts of indeterminacy and choices to be made.