>> "This seems to have been due to the user agent not being set, which I fixed by setting it to "Mozilla/5.0"."
Sorry, this was a configuration error in our DDoS protection. Leaving the useragent blank should work again.
>> "Possibly related: some of my customers reported connection problems today - they can't activate the software."
I just wrote a blog post about this: https://wyday.com/blog/2016/ddos-mitigation-and-expanding-our-servers/
Specifically, customers DNS entries will update automatically. But if their DNS servers are especially slow it could take up to 24 hours to update to the latest correct information (so about 11 more hours from when I'm writing this).
Once the customers have the latest DNS entries, have the Austrailian / New Zealand customers that were having problems connecting before try again. Our DDoS counter-measures may have also solved these customers problems.