Thanks for the advice! Sorry it took me so long to reply.
OK. I've tried a number of things. First, I made sure my WineTricks was up to date. Then I tried the command: "winetricks winhttp" as you suggested. No luck. Same problem.
Bit Che appears to detect Wine. Wine appears in the status bar on start up.
I tried uninstalling Wineskin Winery and reinstalling. No luck. I uninstalled it again.
I tried installing Wine Bottler and Bit Che. No luck. I uninstalled again.
I even tried installing a demo version of Crossover with Bit Che. No luck.
I have also tried turning off the Mac firewall.
I have made sure that Bit Che and Wine have firewall permissions.
I'm a bit baffled. When I click "About" Bit Che, or "Check for version update", or "Check for driver updates" (or whatever the exact text it. I think you know what I mean) There appears to be a brief burst of network traffic, and then a pause of several seconds, followed by another brief burst. It goes on like this until Bit Che times out on the connection.
It's strange, because I have another Wine wrapper running a Windows application. It's called VZ Connect and it's a freeware VOIP voicemail operator program. It's not giving me any problems with Wine or OS X El Capitan. So I'll kee trying and let you know how it goes.
Any additional suggestions would be greatly appreciated. Thanks!