Is it constitutional to outlaw collective bargaining for public employees?

There’s a lot involved in this question, but I’ll try to distill it down as much as I can. Maybe in the form of questions which I hope are not too leading.

Is making it illegal for people to organize themselves into a cooperative group a restriction of free speech?

If it is constitutional to outlaw union organizing, then what do you do when everyone stops working on the same day? What is the penalty? Do you arrest only the leaders? Do leaders even matter in a time with the internet and cell phones; when flash mobs and protests can be organized, without leaders, in a few hours?

What if everyone quits on the same day? Can you force them to work? If you force them to work, can you force them to do a good job? How does this work without being slavery?

Is the attempt to eliminate collective bargaining a credible threat? Or is it a bargaining chip designed to scare people who may not know their rights? Could you please explain to me what the Governors in Wisconsin, Ohio, Indiana, Tennessee and Florida (among others) are thinking?