I feel very disappointed. That's all I have to say...
Of course installing 3rd party browsers in WINE won't make any difference. If you can install/browse with them, that's because they will use 3rd party libraries like OpenSSL and have their own certificate stores which have no dependences on the underlying APIs provided by the OS.I only try to used it to open links, didn't expect that solved the conection issues
All internet connectivity in JScript Panel and Spider Monkey Panel is through the xmlhttp ActiveXObject / built in windows certificate store which is entirely dependent on how updated the OS is. It doesn't necessarily have to be that way but that's the way all scripts are currently written. No one has gone out of their way with alternatives because no one cares about WINE enough.
I suspect this would be servers being updated to block ye olde clients without modern TLS support. It's like Windows 7 is pretty useless out of the box until it's fully updated.