First that comes to mind is New York state (where I live), where the Taylor Law makes it illegal:
"One of the most controversial parts of the Taylor Law is Section 210, which prohibits New York state public employees from striking."
https://en.wikipedia.org/wiki/Taylor_Law
Example of legal punishments handed out a few years ago when transit workers went on strike in NYC:
http://www.labornotes.org/2012/01/public-employees-need-right-strike
Other states are not quite so codified; but here's a ruling in California that establishes "essential" workers cannot strike, including police & firefighters, possibly nurses and teachers:
http://www.slotelaw.com/articles/public-employees-right-strike-clarified-california-supreme-court