Andrew,
Unsure why the error is showing up there. That button is using Joomla's own Update manager to check if an update is available, so that means the error *could* be with Joomla making the outgoing connection to check our XML update file.
Can you go to the Joomla -> Extension Manager -> Updates area and a "Find Updates" from there? If found, you should be able to install the update from there without having to use our admin interface.
That will narrow down the cause of the issue quite a bit if it works or not.
Thanks,
Alex