What is the equivalent to ng-if in react.js?

How about ternary operator?

render() {
  return (
    this.props.showMe ? <button type="submit" className="btn nav-btn-red">SIGN UP</button> : null
  );
}

You can also use &&:

render() {
  return (
    this.props.showMe && <button type="submit" className="btn nav-btn-red">SIGN UP</button>
  );
}

Large block as in the comment can easily be handled by wrapping the JSX in ()s:

render() {
  return this.props.showMe && (
    <div className="container">
      <button type="submit" className="btn nav-btn-red">
        SIGN UP
      </button>
    </div>
  );
}

Also inline:

render() {
  return (
    <div className="container">
      {this.props.showMe && <button type="submit" className="btn nav-btn-red">SIGN UP</button>}
    </div>
  );
}