Many government positions do not give you a right to strike.

The idea of striking is to force the company to raise prices or cut management salary and give more to workers, so it's a type of political battle over the economic surplus created by the firm. That makes sense in, say, a factory or business where there is a surplus to be fought over.

But Government agencies have no surpluses, they run deficits and they have management salaries set by law, so things are pretty much take it or leave it.

On the other hand, workers in these companies benefit from a host of legal protections and legally encoded pay scales so the usual method is to sue that some worker protection was being violated or that a promotion to a higher pay bracket was unfairly withheld. Private sector workers tend not to have these types of regulated interactions with employers and so for them striking is a more appropriate option.

Where you see government unions more is in local government, so say teachers can threaten to go on strike unless the city council raises taxes or cuts some other program to divert more to them. That obviously has disadvantages as it means the less unionized budgets get less funding and its hard for smaller departments to defend their budgets vis-a-vis departments with more union members. Federal government workers, OTOH, which must be funded by Congress and signed by the Executive, are in a different boat as they have less power to threaten Congress than a local government. In either case, the unions are not striking against the management of the agency, so it is not a typical strike. They are striking against the funding authority, so it's not the classical case of a class struggle over the producer surplus. Rather, it's a struggle between workers and taxpayers, not workers and management. The private sector equivalent would be a firm refusing to produce goods if consumers didn't buy them. If that firm was a monopoly. And consumers were legally forced to buy the goods via taxes. Not at all the same thing!

In terms of is there a right to strike, well anyone can quit or refuse to work whenever they want. And they can even organize their coworkers to quit en masse. That's not in dispute. The dispute is whether the company can fire them for not working. Saying that a worker has a "right" to strike is a very oblique way of saying that a company does not have a right to fire a worker that doesn't show up to work.

Now, it's not at all clear why the government, especially critical agencies in the government that provide security for the nation, shouldn't have this right.

Imagine if a fireman said they wouldn't put out a fire unless you paid him more. Wouldn't a normal person respond by saying "Why did you agree to take the job? Why not quit and let someone else, who is willing to do the job, do it? Why hold me hostage?" Or a policeman going on strike and refusing to respond to a call until the citizens of the town voted to increase police pay. Well the money that pays government workers is money from citizens, not some economic profit created by the production process, so demanding more of this money and preventing the agencies from operating until more is voted by the legislature is a pretty unpopular stance except in the most pro-union sliver of the electorate.