Yes. You can modify the index page created by the Loginrequired plugin according to your likings. You would need to modify the files mod/loginrequired/index.php and mod/loginrequired/views/default/page/layouts/loginrequired_index.php to change the content. As a starting point: the content of the register page of Elgg is provided by the code in the file pages/account/register.php. This code would very likely need to be put into mod/loginrequired/index.php or rather replace the present content of this file to make the index page equal a register page. Depending on your likings you might need to make other adjustments, too.

Thanks and sorry for the trouble.