These are the Satmar, one group of Hasidic Jews - as distinguished from the Lubavitchers, the Bobovers, the Skverers and other Hasidic groups.
They are arguably the largest group, and their official teaching is that the State of Israel should not exist.
While this does not stop them from taking enormous amounts of public assistance from the government, they object to serving in the armed forces of that same government.
There is a more extreme faction that is not part of the Satmar but has members who have been affiliated with it called the Neturei Karta - this group actively collaborates with Hamas to destroy the State of Israel.
It is a longstanding theological dispute, at the heart of which is the belief that only the Moshiach (Messiah) can restore Jewish government in Israel, that secular government is an abomination.
Among Hasidim, mileage varies on this theory.
OK, so the Israeli government could just tell them: have it your own way. While you can live here, you are responsible for your own security -- Israeli army elements will no longer protect your community, plus no more public assistance for you.
I thought the other groups were also to varying degrees anti-Zionist. Though nothing is terribly clear.
The best way of looking at this is probably at surface value, that they oppose the draft because they are caught up in a sectarian ghetto mentality, and know that if their children see other situations (and “the big city”), they won’t want “to go back to the farm.”
But it’s a losing argument from two perspectives. The first is that having a military and keeping order are the two primary functions of government. If they don’t support the military, they have reneged on the social contract.
The other reason is that yes, while there will be attrition of some of their children, others will return, and likely with ideas that will revitalize their communities.
And the Neturei carta went to Iran when Ahmadinejad had his Holocaust denial forum. But most Hasids in Israel benefit from this policy that the government wants to change. I realize that part of this is that the Satmars have an anti-Zionist stance—but yet they are benefitting from the Zionist state.