From their FAQ:
“We do not currently have the functionality to delete one’s account.
However, you do have the option to Deactivate your account, which can be found in “My Settings.” Deactivating your account will make your profile page and broadcasts disappear from public view. They will not be found via direct links, search or reruns. However, some information may still be visible to others, such as notifications for gifts you’ve given on YouNow, or private messages you’ve sent.