This is a Premium feature of Cowlendar. You need to upgrade your plan if you want to use this feature and you are a free user of Cowlendar.
This is perfect for when you are going on vacation or even have to close a date out last minute to deal with an emergency
Steps for blocking the date on a specific service
In your Shopify admin dashboard look at the left panel and click on Apps.
Click on Cowlendar.
Click on Settings.
Scroll down a bit and find Add a blocked date.
Bonus : you can block a specific date. In the picture below i've blocked January 10th to January 10th from 00h to 24h meaning the entire day won't be bookable.
If there is a greyed out date this means it has already passed and cannot be edited (but can be deleted)