My issue has now been resolved.
The python version of my buildout was updated some time ago. I had to reinstall modules that used C extensions and needed to be recompiled for the new Python version. Most of these modules caused an error during startup so it was easy to know which ones needed to be removed so buildout can download and recompile them.
The zeoserver log showed that zodbpickle was causing the error so I reviewed its package description on PYPI and upon reading that it uses C extensions, I knew that I had to remove it. After it was reinstalled, zeopack was able to proceed without problems.
This issue could be related to another problem that I reported here (Missing.missing error)