Now it works!

index.ios.js

import React, { AppRegistry, Component, Navigator, View, Text, TouchableHighlight } from 'react-native'
import { Router, Route, Actions } from 'react-native-router-flux'

class First extends Component {
  render() {
    return (
      <View style={{flex: 1, alignItems: 'center', justifyContent: 'center'}}>
        <Text>This is shown when not signed in</Text>
        <TouchableHighlight onPress={Actions.second}>
          <Text>Next</Text>
        </TouchableHighlight>
      </View>
    )
  }
}

class Second extends Component {
  render() {
    return (
      <View style={{flex: 1, alignItems: 'center', justifyContent: 'center'}}>
        <Text>Now we can sign in</Text>
        <TouchableHighlight onPress={this.props.onSignIn}>
          <Text>Sign in</Text>
        </TouchableHighlight>
      </View>
    )
  }
}

class NotSignedIn extends Component {
  componentWillUnmount() {
    Actions.currentRouter = null
  }

  render() {
    return (
      <Router>
        <Route {...this.props} name="first" component={First} initial={true} title="First" />
        <Route {...this.props} name="second" component={Second} title="Second" />
      </Router>
    )
  }
}

class SignedIn extends Component {
  render() {
    return (
      <View style={{flex: 1, alignItems: 'center', justifyContent: 'center'}}>
        <Text>This is shown when signed in</Text>
        <TouchableHighlight onPress={this.props.onSignOut}>
          <Text>Sign out</Text>
        </TouchableHighlight>
      </View>
    )
  }
}

class Root extends Component {
  constructor(props, context) {
    super(props, context)
    this.state = {
      signedIn: false,
    }
  }

  handleSignIn() {
    this.setState({signedIn: true})
  }

  handleSignOut() {
    this.setState({signedIn: false})
  }

  render() {
    const { signedIn } = this.state
    if (signedIn)
      return (<SignedIn onSignOut={this.handleSignOut.bind(this)} />)
    return (<NotSignedIn onSignIn={this.handleSignIn.bind(this)} />)
  }
}

AppRegistry.registerComponent('RoutesTest', () => Root)