These two functions create anchor links, and you can customize a lot of it, but it’s impossible to add a class through just using their parameters alone. Gotta add a function to functions.php:

add_filter('next_posts_link_attributes', 'posts_link_attributes');
add_filter('previous_posts_link_attributes', 'posts_link_attributes');

function posts_link_attributes() {
    return 'class="styled-button"';
}