You're talking about the black, square "blocks" which the two pigtails attach to on either side? Then yes, you'll have to get that resoldered. I can see why they would detach it, though I don't know why they wouldn't just have cut a little square for the opening and slid the pigtail through the hole. Sure, it might have required two people, but it would have looked better.
From what I'm reading, it shouldn't be an issue to tint over defroster lines and still use the defroster (some people wait a bit before using it, though) as long as they are a good installer.
As a side note, you didn't notice the pigtails just dangling there when you check the liftgate, or on the ride home? Or you didn't check the tint before paying?
Either way, I hope you get it resolved soon, and hopefully they won't give you grief because you've already signed, paid, and "accepted" the job as complete.