React: trigger onChange if input value is changing by state?

You need to trigger the onChange event manually. On text inputs onChange listens for input events.

So in you handleClick function you need to trigger event like

handleClick () {
    this.setState({value: 'another random text'})
    var event = new Event('input', { bubbles: true });
    this.myinput.dispatchEvent(event);
  }

Complete code

class App extends React.Component {
  constructor(props) {
    super(props)
    this.state = {
    value: 'random text'
    }
  }
  handleChange (e) {
    console.log('handle change called')
  }
  handleClick () {
    this.setState({value: 'another random text'})
    var event = new Event('input', { bubbles: true });
    this.myinput.dispatchEvent(event);
  }
  render () {
    return (
      <div>
        <input readOnly value={this.state.value} onChange={(e) => {this.handleChange(e)}} ref={(input)=> this.myinput = input}/>
        <button onClick={this.handleClick.bind(this)}>Change Input</button>
      </div>
    )
  }
}

ReactDOM.render(<App />,  document.getElementById('app'))

Codepen

Edit:
As Suggested by @Samuel in the comments, a simpler way would be to call handleChange from handleClick if you don't need to the event object in handleChange like

handleClick () {
    this.setState({value: 'another random text'})
    this.handleChange();
  }

I hope this is what you need and it helps you.