The IAM cannot shut down the airline and is bound to work by their CBA. If the union or its members conduct "wild cat" job actions, the company would likely go to court to obtain a court order forcing the mechanics, utility and fleet service personnel to go to work.
If a member violates a company policy, then the mechanic will be terminated, which you may begin to see shortly. Furthermore, I expect to see more mechanic furloughs in addition to those who will lose their job because Pittsburgh de-icing and baggage service maintenance will now be conducted by the ACAA.
From a historical perspective, in the 90's APA tried an illegal sick out over the Reno Air - AMR integration and received a fine for $45 million, which could have been accessed on the membership.
In the case of the IAM and US Airways, the choice is simple: Either the mechanics negotiate a cost effective way to overhaul aircraft (provided they lose the A320 heavy maintenance grievance as expected) or the company ships the work out and virtually all of the mechanics lose their job.
It's pretty simple and from this observer's perch, the mechanics have very little leverage.
Regards,
USA320Pilot