You could create a form with a hidden field that contains the uid, another one that contains another hidden variable and the button leads to your logout.php something like this:
<form method="post" action="logout.php">
<input type="hidden" name="uid" value="<?=$uid; ?>" />
<input type="hidden" name="checked" value="checked" />
<input type="submit" name="Logout" value="Logout" />
</form>
Then on your logout.php page you get your post variables
$uid = $_POST["uid"];
$check = $_POST["checked"];
And now you have both the uid that you wanted and a variable that contains "checked" when the form is submitted.
This is a quite rudimentary way of doing things, but it will work. You can either pass the variables through standard php means or ajax, either way this should work.