In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.