No, you don't need to do anything special on your OAuth endpoint. (It has to work, of course, but it isn't tested by the health checker.)
There are many reasons why your webhook may be judged as "unhealthy", even if you think you're handling it correctly. The most likely, of course, is that you're not - you may be throwing an error when you don't think you are. If you are not handling the health check specifically, you may also be using features that the health checker does not "support" (for example, sending back a Media response).
Checking logs, both for your webhook and in Cloudwatch for the Assistant, may help you narrow your problem down. If you update your question, we may be able to provide additional guidance.