Update: I've check my firewall, security software, router, disabled UDP Checksum Offload and reset my DNS cache and nothing resolved the problem.
I then went into TeamSpeak itself, accessed its global list of available servers, and was able to successfully connect to random servers in Sweden, France, and Canada, all with the name ending in ".teamspeak.network". I've even been able to connect to random servers throughout the US, the UK, Japan, and Saudi Arabia.
So the connectivity problem really doesn't seem to have anything to do with my firewall, security software, or my router.
However, the TeamSpeak server list search function cannot find any server name starting with or containing "rarebreed". I'm at my wit's end. Any further thoughts?