add the checkbox to the checkout page.
also add a hidden form element with the same name to the confirm page:
to checkout:
<input type="checkbox" name="nextDay" value="1" />
to confirm:
<input type="hidden" name="nextDay" value="<?php echo isset($_POST['nextDay']?$_POST['nextDay']:""); ?>" />
On the confirm page, add a set session value server behavior. On the Server behavior panel, click the plus button and select eCart -> General -> Set Session Value.
For both the trigger and the value, click the lightning bolt and select the nextDay form element.
set the name of the session variable to "nextDay"
now create a new shipping rule using the Session Variable exists "nextDay" trigger and based on session variable value trigger:
if the session variable "nextDay" is "=" to a value of "1"
and set a flat rate charge of 5.75