How to insert a Button or custom text on twentyseventeen header

I’m guessing in involves PHP to the site-branding file, but not really sure.

yes it does. You to create a child theme based on the twentyseventeen theme, and add create a new custom header.php file with the modifications you need, in order to style your header. Your custom twenty seventeen header file would look something like,

<?php
/**
 * The header for our theme
 *
 * This is the template that displays all of the <head> section and everything up until <div id="content">
 *
 * @link https://developer.wordpress.org/themes/basics/template-files/#template-partials
 *
 * @package WordPress
 * @subpackage Twenty_Seventeen
 * @since 1.0
 * @version 1.0
 */

?><!DOCTYPE html>
<html <?php language_attributes(); ?> class="no-js no-svg">
<head>
<meta charset="<?php bloginfo( 'charset' ); ?>">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="profile" href="http://gmpg.org/xfn/11">

<?php wp_head(); ?>
</head>

<body <?php body_class(); ?>>
<div id="page" class="site">
    <a class="skip-link screen-reader-text" href="#content"><?php _e( 'Skip to content', 'twentyseventeen' ); ?></a>

    <header id="masthead" class="site-header" role="banner">

        <?php get_template_part( 'template-parts/header/header', 'image' ); ?>

        <?php if ( has_nav_menu( 'top' ) ) : ?>
            <div class="navigation-top">
                <div class="wrap">
                    <?php get_template_part( 'template-parts/navigation/navigation', 'top' ); ?>
                </div><!-- .wrap -->
            </div><!-- .navigation-top -->
        <?php endif; ?>
        <button>
         <!-- your custom button -->
        </button>

    </header><!-- #masthead -->

    <?php

    /*
     * If a regular post or page, and not the front page, show the featured image.
     * Using get_queried_object_id() here since the $post global may not be set before a call to the_post().
     */
    if ( ( is_single() || ( is_page() && ! twentyseventeen_is_frontpage() ) ) && has_post_thumbnail( get_queried_object_id() ) ) :
        echo '<div class="single-featured-image-header">';
        echo get_the_post_thumbnail( get_queried_object_id(), 'twentyseventeen-featured-image' );
        echo '</div><!-- .single-featured-image-header -->';
    endif;
    ?>

    <div class="site-content-contain">
        <div id="content" class="site-content">

Notice the place where I have inserted a custom button with the html comment <!-- your custom button -->