Great, thanks Rui. That whited the screen for me at least, but copying the Login section and adding Sign Up with the below code worked.
} else {
$aux_links_output .= '<li><a href="'.$login_url.'">'. __("Log In", "swiftframework") .'</a></li>'. "\n";
$aux_links_output .= '<li><a href="'.$login_url.'">'. __("Sign Up", "swiftframework") .'</a></li>'. "\n";
}
For other readers, you can then style this 2nd 'Sign Up' child link with:
.top-header-menu .menu > li:nth-child(2) > a {
background-color: #0274be;
text-align: center !important;
border-radius: 5px;
color: #ffffff;
font-weight: 500;
margin-top: 10px !important;
min-width: 66px !important;
padding: 1px 12px 3px !important;
margin-left: 4px !important;
}`