Private sector unions are crap under US law, but they're a good concept in general.

No, public sector employees should not be allowed to strike. They get special protections (civil service) that do not apply to private-sector workers. Public sector workers work for the rest of us; we're not all fat-cat capitalists taking their potential wages to make ourselves rich. We're their fellow citizens, and when government employees strike, what's the average citizen to do? Your mother died while abroad, your passport is expired, but hey, the State Department clerks are on strike, so guess what? You're not going.

You want things to change, vote someone in who agrees with you like any other citizen. The only strike-breaking measure that's easily available to the government (due to civil-service laws) is conscripting everyone in the job or replacing them with military/National Guard, which is swatting a fly with a Buick.