The author establishes a conditional weighted estimate on \(L^1\) nontangential maximal functions in terms of a suitable square function of solutions to an elliptic second-order homogeneous linear differential equation in a Lipschitz domain, assuming that one already has an estimate on the non-tangential maximal function on this domain in terms of the square function. As a consequence, the author obtains exponential square integrability estimates of John-Nirenberg type for the boundary values of harmonic functions in a Lipschitz domain for which the square function is bounded.