config-overrides.js

const findLoader = (config, test) => {
  const loaders = [].concat(config.module.loaders);
  while (loaders.length) {
    const loader = loaders.shift()
    if (loader.test && loader.test.toString() === test.toString()) {
      return { loader, index: loaders.length }
    }
  }
}
module.exports = function override(config, env) {
  const { loader, index } = findLoader(config, /\.css$/)
  loader.loader = [
    'style',
    'css?modules=1&importLoaders=1',
    'postcss'
  ].join('!')
  loader.exclude = /src\/styles\/index\.css/
  config.module.loaders[index] = loader
  return config;
}

package.json

{
  "name": "movieApp",
  "version": "0.1.0",
  "private": true,
  "devDependencies": {
    "autoprefixer": "^6.7.7",
    "babel-plugin-transform-postcss": "^0.1.2",
    "npm-run-all": "^4.0.2",
    "postcss": "^5.2.16",
    "postcss-cli": "^3.0.0-beta",
    "postcss-import": "^9.1.0",
    "postcss-modules": "^0.6.4",
    "precss": "^1.4.0",
    "react-app-rewired": "^0.6.5",
    "react-scripts": "0.9.5"
  },
  "dependencies": {
    "babel": "^6.23.0",
    "react": "^15.4.2",
    "react-dom": "^15.4.2"
  },
  "scripts": {
    "start:react": "react-app-rewired start",
    "build:react": "react-app-rewired build",
    "build:styles": "./node_modules/.bin/postcss --config postcss.config.js src/styles/index.css -o src/index.css",
    "start:styles": "npm run build:styles -- --watch",
    "start": "npm-run-all --parallel start:*",
    "build": "npm-run-all build:*",
    "test": "react-scripts test --env=jsdom",
    "eject": "react-scripts eject"
  }
}

postcss.config.js

module.exports = ctx => {
  return {
    plugins: [
      require("postcss-import")(),
      require("precss")(),
      require("autoprefixer")({
        "browsers": "> 5%",
      })
    ]
  }
}