wp-login.php register/lost password links to buttons?

There’s no way to change the output of the file because the there aren’t any filters you can hook into.

However, there are some hooks that you can use like the login_enqueue_scripts hook to enqueue jQuery and/or a custom javascript. And then login_footer to print some jQuery that will add classes that you can then style with CSS.

add_action( 'login_enqueue_scripts', 'wpse_login_enqueue_scripts' );
function wpse_login_enqueue_scripts() {
  wp_enqueue_script( 'jquery' );
}
add_action( 'login_footer', 'wpse_login_footer' );
function wpse_login_footer() {
  print( '<script>jQuery("a[href$=register]").addClass("button");'
  . 'jQuery("a[href$=lostpassword]").addClass("button");</script>'
  . '<style>.login #nav a.button{color:#123456;}</style> );
}

This code would go either in your functions.php file or a separate plugin.