Question: I have experience testing from a business perspective but I'm currently working in an system which is a small part of a bigger system, so I don't have a clear vision of the entire business value flow.

From a black box perspective its just a service with one input and one output:

Input: a cronjob starts the service and it will query a DB for specific cases where it should send an email;

Output: the system sends a reminder email and write in the DB that the email was sent.

The developers write unitary tests, integration tests and system test. Its pretty well covered by tests from a white box perspective.

From a black box perspective, which kind of test can I put in place?

Answer: 
I would start with what you have, the inputs and outputs, then start questioning. I like to go through a bunch of what if's, such as in your case "what if the database is unavailable?" or "what if the mail queue is full?" or something like that. If you can't come up with questions like that, then ask the developers "what didn't you test that you think I should?".

Often I am testing a small portion of a larger system in isolation, or just changes to only a specific smaller piece of something larger. Then you need to check the overall flows for the bigger pieces to the smaller, and the other way around. Definitely start with the requirements, and there are always requirements somewhere, and then start looking at what you have and think outside the box, so to speak, to whatever else there is that might fail or be an input or output that may or may not fail, or be other than what was expected.

I find it helps to know the whole system, but its often not necessary, as in many cases it depends on what it is you are trying to accomplish.
Your test cases have to be based on the requirements for your system.

If no requirements have been documented, you should ask some people from the project or from the appropriate business department for the requirements.

In your case, these requirements would probably contain the specific cases when an e-mail has to be sent, what content and which receiver the e-mail should have and so forth.

You do not need to know the entire business flow to test your system.