Magento 2 : add custom class on checkout page shipping address field

You can create plugin after Attribute merge method. Then you can modify checkout fields and add class to wrapper of country select box.

In your module create MODULE/NAME/etc/di.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="../../../../../vendor/magento/framework/ObjectManager/etc/config.xsd"> 
    <type name="\Magento\Checkout\Block\Checkout\AttributeMerger">
        <plugin name="customAttributeMerger" type="\MODULE\NAME\Model\Plugin\AttributeMergerPlugin"/>
    </type>
</config>

Then in \MODULE\NAME\Model\Plugin\AttributeMergerPlugin.php

<?php

namespace MODULE\NAME\Model\Plugin;

class AttributeMergerPlugin
{
    public function afterMerge(\Magento\Checkout\Block\Checkout\AttributeMerger $subject, $result)
    {
        if (array_key_exists('country_id', $result)) {
            $result['country_id']['additionalClasses'] = 'your_custom_class';
        }

        return $result;
    }
}

For safety clear cache and you should see this class in div wrapper of country select.

Then you can use .your_custom_class select in css/less or $('.your_custom_class').find('select'); in jQuery. Remember that JS script you have to execute when fields are generated