Using third party subscriptions
First off, it is possible to support an existing subscription system in your app, but it must be in addition to the App Store subscriptions. Also you have to be very careful about the UI you use. You can't prompt users to bypass the App Store and purchase through your server, but you can offer existing subscribers the ability to sign in with credentials they may have previously obtained through your website.
That said, Apple can be fairly capricious in their rulings on what is and isn't allowed in the App Store. I have worked on apps that have done this sort of thing, but they have been for fairly well known magazine publications. There's no guarantee that they will allow it for everyone.
Detecting unsubscribed App Store subscriptions
As for using App Store subscriptions, when a user purchases a subscription in your app, they will receive a receipt in the SKPaymentTransaction object. This receipt should be posted to your backend server to make sure it is valid before you give the user access to anything. See the In-App Purchase Programming Guide for more details.
When you setup a subscription type, you specify how long that subscription lasts. So if you log the transaction date in the SKPaymentTransaction when you receive a receipt, you can determine exactly when that subscription should expire by adding the duration of the subscription to the transaction date.
If it's an auto-renewing subscription, you will receive a new receipt when the subscription is renewed. So once you have validated that with your backend server, you can update your expiry date based on the new transaction date. If you don't receive a new receipt before the first one expires, it's likely the user has cancelled their subscription.
On the backend, your server can also tell when a particular subscription will expire based on the response from the verification server. First there is the status code which will tell you whether the subscription has already expired, but there is also an expiry date returned in the decoded receipt which will tell you when it is expected to expire if it hasn't already.
For more details, see the Auto-Renewable Subscriptions documentation.
Testing auto-renewing subscriptions
It's worth noting that when you are testing auto-renewing subscriptions in the App Store sandbox environment, the length of the various subscription types is dramatically shortened to make it easier to test. For example a 1 week subscription lasts only 3 minutes in the sandbox environment. You can see the full list of times in the iTunes Connect Developer Guide.