As canterburry says above, good communication can go a long way towards improving customer experience and avoiding problems.

If you can provide clients with ample warning about any upcoming changes, and preferably provide them with a staging/dev instance they can test their code against, it might help avoid similar problems in future.

In my experience, clients get frustrated when their business-critical processes suddenly break with no warning and they have to engage in fire-fighting. From their perspective, they had something working one day, then you "made a change" and "all their stuff broke". That might not be a fair assessment, and it might well be due to shoddy practices on their end, but the end result is the same: the client is frustrated and has a bad experience of using your product.

It becomes especially tricky when any given customer can hard code an SSL cert into the Websphere or Tomcat server without the service provider knowing anything about it. Then, when the service provider needs to rotate the certificate because it's about to expire and is renewed or perhaps it's suddenly rotated (because of Heartbleed, for example), it breaks the customer.

What kind of communication channels have you guys seen or used for that kind of scenario? A separate Twitter feed? A blog? Sending an email to all customer every time the SSL certificate changes in any way?

More or less any communication channel will work, provided clients are aware of it and checking it regularly. Personally I would recommend a mailing list that your clients can subscribe to in order to be notified of any upcoming changes.