As I no longer play this game I can't keep up what's broken and not. The solution below is for when the game launched first 6 months. After many iterations and new specializations added the game may bug this may unlock early or not at all for you. I'd advise to read comments below this to see how some people are unlocking it..
After completing the last Stronghold main story mission 'Capitol Hill' you will have access to specializations.
At the White House there is a table next to the quartermaster where you can choose between demolitionist with a grenade launcher, sharpshooter with a .50 Cal and survivalist with a crossbow. Once you choose your expertise of choice you can just run out the White House and kill the first mob you see with your specialist weapon. Hold Y button to switch to it.
Once you get a kill with either specialist weapon return to the White House and activate another specialist, rinse and repeat 2 more times.