variable created in page.php is null inside of header.php

Generally said, there should nothing be happening before the get_header() call.

So initiate your class if is_page() is true via wp_head action hook. Example:

add_action( 'wp_head', 'initiate_my_class' );
function initiate_my_class() {
    if ( is_page() ) {
        //code
    }
}

Edit: As @AndrewBartel in his comment said you can globalize your variable or implement some kind of storage system via the class or another one.