React - how to pass state to another component

Move all of your state and your handleClick function from Header to your MainWrapper component.

Then pass values as props to all components that need to share this functionality.

class MainWrapper extends React.Component {
    constructor() {
        super();
        this.state = {
            sidbarPushCollapsed: false,
            profileCollapsed: false
        };
        this.handleClick = this.handleClick.bind(this);
    }
    handleClick() {
        this.setState({
            sidbarPushCollapsed: !this.state.sidbarPushCollapsed,
            profileCollapsed: !this.state.profileCollapsed

        });
    }
    render() {
        return (
           //...
           <Header 
               handleClick={this.handleClick} 
               sidbarPushCollapsed={this.state.sidbarPushCollapsed}
               profileCollapsed={this.state.profileCollapsed} />
        );

Then in your Header's render() method, you'd use this.props:

<button type="button" id="sidbarPush" onClick={this.props.handleClick} profile={this.props.profileCollapsed}>