Yes, Ethereum should be usable in the same way that Quora or some other private chain system would, provided you use an alternative to proof-of-work, like Parity's proof-of-authority implementation. Doing this has the benefit that any software you build around your private chain should also be usable on the public chain, should you need to transfer it.
However, depending on the number of users and the relationships between them, the following requirement may be problematic, whether you're using Ethereum or some other blockchain system:
The transactions need not be totally private i.e. it's not an issue if the transactions of a user are seen by other users. But my network itself should be secure and private i.e. not available for public users unless they are registered.
If your users interact with the blockchain the normal way, any user who has permission can see all the transactions of all the other users. This is OK if there's a small number of users and they are all trusted, but if you're dealing with large numbers of registered users, you may not be able to trust them to protect their credentials and prevent non-registered users from seeing it.
If the risk of leakage is unacceptable, you may need to put an additional server-side layer between your users and the blockchain, so the users don't actually get to access the blockchain directly and can only see the transactions that affect them directly. This reduces the risk of leakage to unregistered parties, but it may in turn conflict with whatever requirement is prompting you to use a blockchain for this project.
