Forgot to tell you how to automatically update the Country so there is no need for the update button and will save your customers some time!

Open the following file:
includes\templates\YOUR TEMPLATE\templates\tpl_modules_shipping_estimator.php

Find the following code
Code:
<label class="inputLabel" for="country"><?php echo ENTRY_COUNTRY; ?></label>
<?php echo zen_get_country_list('zone_country_id', $selected_country, 'id="country" onchange="update_zone(this.form);"'); ?>
<br class="clearBoth" />
Replace that code with the following code
Code:
<label class="inputLabel" for="country"><?php echo ENTRY_COUNTRY; ?></label>
<?php echo zen_get_country_list('zone_country_id', $selected_country, 'id="country" onChange="this.form.submit();"'); ?>
<br class="clearBoth" />
Remove the following code
Code:
<div class="buttonRow forward"><a href="_" onclick="return shipincart_submit('');"><?php echo  zen_image_button(BUTTON_IMAGE_UPDATE, BUTTON_UPDATE_ALT); ?></a></div>