How does the Eng Repair feature work?

Which icon? On PC pressing X opens up your torch to repair vehicles. If you mean the one on that right side of the screen for the 'super repair' that's a squad bonus

Sometimes it absolutely refuses to whip the repair tool out, seems to be tied to the shield icon in the bottom right corner.

Gotcha, I haven't experienced that, I was splitting my time last night between Support and Engineer and never had it not come out by hitting X (or the supply pack when playing support). Are you on PC or console? Did you rebind anything in settings that may be causing a conflict?