This study presents a high order moving interface treatment for fluid-structure interaction (FSI) problems between compressible viscous flows and deformable structure. A loosely partitioned coupling strategy that involves flow solver, structure solver and fluid-structure interface treatment is considered. In particular, the compressible Navier-Stokes equations are solved by a high order finite difference method on fixed Cartesian Eulerian grids. The structure, governed by linear elastic dynamic equation, is solved by a finite element method in Lagrangian framework. In terms of the fluid-structure interaction interface treatment, a new simplified inverse Lax-Wendroff (ILW) approach is extended and implemented to ensure high order accuracy. Several numerical tests are provided, including accuracy test, shock-loaded cylinder and deformation of a panel induced by shock waves. The numerical results demonstrate that the proposed method is capable of handling FSI problems involving shock wave and structure deformations non-oscillatorily and with high order accuracy for smooth solutions.