I tried your suggestion, but it had no effect. I tried disabling all virus software and firewalls, but that didn’t help. The internet connection was fine throughout, and I can SSH into the Pi and ping various external IP addresses.
A new problem is that the Pi now won’t automatically mount the external drive, even though I’ve added it to fstab. However, now that I’ve mounted the drive, the Squeezebox server is running three clients successfully, so at least I’ve cured the original problem (long periods of silence from all the Squeezebox clients). For now at least.