Hacking into State institution computer systems should the Governments make use of the hacker's talents?

This is not a question.

Did you see Catch Me If You Can? The government already does this type of recruitment.

The government does use hackers, Feds Seeking Computer Hackers to Secure Nation’s Networks.

Is this a question or a job offer?

Yes, as others have pointed out, they already do. But hackers are criminals, some better intentioned than others I suppose, perhaps politically motivated as opposed to being criminally motivated, or they just do it because they can, but they are still criminals. You have to choose your criminals wisely. Not all of them are suitable candidates for law enforcement.

The government does use hackers, but they are also known to waste/misuse resources; two things that any true hacker detests and abhors, so many only really work there grudgingly.

The smart hackers often get caught, do a little time, put that on their resume, and get a lucrative job in the private sector. Trading a few months of freedom for a career with a six-figure income is a fair trade in the minds of some of them, and its better to work for an employer that knows the difference between 2012 and 1962, (Government really hasn’t grasped the whole Computer Age thing well enough to figure out how to deal with cyberspace.)