A couple of ideas:
“my host recently upgraded PHP to 5.6.” – well, this really isn’t much of an upgrade as any decent hosting company is now using PHP 7.
I do suspect some sort of PHP compatibility issue.
Note also that you need to have the latest Weaver Xtreme 3.2.3 version as well.
But the real key here is:
“More information about this error may be available in the server error log.”
That is most likely to have an entry saying exactly where things are crashing, and you can access the error log from your host control panel.
But we’ve not had reports of any issue with the latest support plugin, so some sort of server error is most likely.