"Like a graveyard...

More fun with about:config

Here's the key name you want to find: "browser.tabs.closeButtons" and you want to set it to "2" if you want to kill all the close buttons. If you set it to "0", you will find that only the active tab has the close button.

For me, my close button is the MIDDLE MOUSE BUTTON and I don't want screen real estate taken up.

Put in your .screenrc file the following line:
termcap linux XT:true
Which tells screen that any terminal identifying itself as "linux" (which is the default setting for PuTTY) has XTerm capabilities. If your system uses terminfo instead of termcap, you can use the "terminfo" command instead. Or even better, you (and probably I [actually, that's what I'm doing now]) could use "termcapinfo" which covers all your bases so you may best move your zig.