Since the problem was specific to certain podcast downloads that use HTTPS to deliver the MP3, and the problem was happening on more than one computer, I figured it was not due to a local problem, or configuration error. I was having even more trouble with the feed
https://mises.org/feed/rss.xml. I was getting a connection error on the feed.
What I have determined is that these feeds are using AES-256 encryption, and do not offer any 128 bit ciphers. Since I am running Windows XP, there are no 256 bit ciphers available in the Windows network stack. I was confused because the links would open in Firefox, but Firefox uses it's own encryption rather than depending on Windows. When I would try the links in Internet Explorer 8, the connection would fail because there were no common ciphers available, but IE did not provide a helpful error message that explained this.
I don't imagine that there is an easy way to fix this in MusicBee, so it will be a growing issue for the XP holdouts as more feeds go to 256 bit only encryption, but I wanted to get the problem documented here in case it will help another user figure out what the problem is.
My solution will be to switch to Linux, hope you can make a MusicBee for Linux some day.