When ShippingEasy notifies Shopify about a shipment, by default, ShippingEasy does not include any directives instructing Shopify, as to whether it should send a shipment notification e-mail to your customers or not. This means that whether Shopify sends a shipment notification e-mail or not, is up to Shopify.
For orders that are placed directly in your Shopify store, Shopify will send a shipment notification e-mail, even though ShippingEasy does not request Shopify to do so. There is no way to disable that Shopify shipment confirmation e-mail.
For orders that were placed indirectly, typically via a Shopify application (for example, something like ReCharge), Shopify does not send a shipment notification e-mail by default.
If you want Shopify to send shipment notification e-mail messages for orders that were placed indirectly, check the box in ShippingEasy for the Request that Shopify send a shipment notification to the buyer: