Try saving the ACL after the user has already been created.
PFUser *user = [PFUser user];
user.username = self.userNameTextField.text;
user.password = self.passwordTextField.text;
user.email = self.emailTextField.text;
[user signUpInBackgroundWithBlock:^(BOOL succeeded, NSError *error) {
if (!error)
{
PFACL *userACL = [PFACL ACLWithUser:[PFUser currentUser]];
[userACL setPublicReadAccess:NO];
[userACL setPublicWriteAccess:NO];
[PFUser currentUser].ACL = userACL;
[[PFUser currentUser] saveInBackground];
} else {
NSString *errorString = [error userInfo][@"error"];
// Show the errorString somewhere and let the user try again.
}
}];