Hi,
Regarding an https connection, I don't think that would work. It might be irrelevant, though. SSH works differently than SSL (which is what HTTPS typically uses). If you connect to a machine via SSH then the connection is encrypted, but it won't be running in an SSL layer. If you have an HTTPS connection then you don't need SSH, and vice versa. Clear as mud?
To pass commands to a unix shell you can and probably should use your puTTY tools. puTTY can open up a secure shell that you can type in commands right at the keyboard. The putty "suite" offers as well psftp.exe which you can use for secure file transfers (probably both at the command line and scripted - if scripted, it is no more than the same commands you would type at the command line, just automated). Not be be outdone, puTTY also gives you pscp.exe which allows you to use scp for secure file transfers. This last is what I tested and provided some code for above.
Bottom line - if you connect to a *nix server you'll have all the SSH tools at your service, and puTTY will give you a windows client to make use of the same protocols. The *nix server must be listening for SSH traffic. By default that's on port 22 but some admins use a different port for ssh traffic.
HTH - it's a bit of trial and error at first. I suggest you attempt to simply open a puTTY shell to see if you can access the server at all (assuming you have a user account to access. Just open puTTY.exe from the start menu and try to open a connection --> username@192.168.1.100)