React: Children onclick to change parent's state for re-rendering

What you need to do is to pass a method from the parent to the child, That it can call when the button is clicked. This method that belongs to the parent will change the state.
In MainPage.js

changeButtonState(event) {
    this.setState({btnNewScreen: !this.state.btnNewScreen})
}

pass this method to your child component as

<Mainpage_Addscreen buttonClick={this.changeButtonState.bind(this)} />

and finally in the child component,

<Button ....   onClick={this.props.buttonClick} />

What you probably need is a callback function, which your parent passes as a prop to your child, and that your child can then call.

Something like this:

// In parent
constructor() {
  ...
  this.onChildClicked = this.onChildClicked.bind(this);
}

onChildClicked() {
  this.setState({childWasClicked : True });
}

render() {
  ...
  return (
    <div>
      <Child onClicked={this.onChildClicked} />
    </div>
  )
}

// In child
render() {
  ...
  return (
    <div>
      <button onClick={this.props.onClicked} />
    </div>
  )
}

PS: I notice that you have a capitalized <Button> component. This is typically for components that you have defined yourself. Standard DOM elements take lowercase, e.g. <div>, <p> etc