Can it disappear after user clicks "closing box"?

Closing box? Not quite sure what you mean.

There is close button in my notification bar and I don’t want to show this bar to the people who clicked this close button.

As long as the user is logged in, they won’t see it again. If you want it to be hidden for guests, there is a pro version with that: https://pippinsplugins.com/simple-notices-pro-plugin-for-wordpress/