You can delete your Hangout account by logging into your account and then click “Delete your account” from Delete a service or your account section.
Here are the steps to delete your Hangout account.
Log into your Hangout account and click on your profile.
Click manage your Google Account from the drop down.
On the left side, click Data & personalisation.
From the Data & personalisation section, click “Delete a service or your account”
Now click Delete your account from the service page.
You have successfully deleted your Hangout account.