To avoid quantile crossings, we propose a nonparametric quantile regression method using a deep neural network with a modified linear unit penalty function. This penalty function is computationally feasible for enforcing non-crossing constraints in multidimensional nonparametric quantile regression. We establish a non-asymptotic upper bound for the excess risk of the proposed nonparametric quantile regression function estimator. The error bars achieve the best minimax convergence rate for the Holder class, and the prefactors of the error bars depend polynomially on the dimensions of the predictors, not exponentially. Construct equiangular prediction intervals that are fully adaptive to heterogeneity based on a proposed noncross-penalized deep quantile regression. The proposed prediction intervals are shown to have good properties in terms of validity and accuracy under reasonable conditions. We also derive a non-asymptotic upper bound on the length difference between the proposed non-crossing conformal prediction interval and the theoretically oracle prediction interval. Numerical experiments including simulation studies and real data examples are performed to demonstrate the effectiveness of the proposed method.