Let me stress that this describes historic events. This does not tell us what will happen in the future. I will have try to have more on this tomorrow. This is from Social Security's official 
history (emphasis added):
On November 14, 1995, SSA was advised to begin a shutdown of operations                 [due to a budget stalemate] beginning at 10 a.m.  The Agency’s contingency plan was implemented.                 Employees were instructed to pay close attention to the media to                determine when the impasse was resolved and when to return to work.                 SSA furloughed 61,415 employees and retained 4,780.  The majority                of the employees retained were in direct service positions to ensure                the continuance of benefits. ...  Other functions considered essential and permitted                under applicable statutes (including the Anti-Deficiency Act) during                a lapse in appropriations related to areas of health and safety,                oversight and coordination of all human resources and financial                functions related to the furlough.             
                           SSA’s contingency plan was altered shortly after the shutdown began                when the Commissioner approved the recall of an additional 130 Office                of Hearings and Appeals employees to handle time-sensitive mail                in the hearing offices.  Without the assistance of these employees,                this claims-related mail would have been returned to the sender,                further delaying claims processing. ...
          It was evident to Agency                executives after a couple of days that the shutdown was not acceptable,                and that it was affecting some of its most vulnerable people, namely                its elderly, disabled, and low-income recipients.  It became                very clear that it was important that SSA reopen its offices to                conduct business, even during the shutdown.  There was tremendous                concern internally, as well as within the White House that SSA not                providing service would cause harm to too many Americans.
             
President Clinton held a press                conference that afternoon and announced to the public that this                shutdown was unacceptable, that Americans needed to be able to conduct                business, and that SSA was recalling approximately 49,715 more employees                in direct service positions back to work.  This involved recalling                sufficient staff to effectively administer its programs, including                processing new claims for Social Security benefits.  This put                field offices, the toll-free 800 number service and State DDS facilities                back in business.  Although SSA would take and process new                applications for benefits and resume processing appeals, it would                not process applications for SSNs, perform personal earnings and                benefits estimates or conduct annual wage reporting activities.                 A total of 54,495 staff would be retained. ...
                          There was a great deal of discussion and some dissention internally                over which employees should return to work, and that if SSA was                going to recall 54,000 employees, why not recall all 66,000.                 The concern was that by recalling all but 12,000 employees, what                message were you sending to them? ...
             
The second shutdown was the longest in history and presented new                challenges to Agency management.  When the threat of a second                shutdown surfaced, SSA determined that the trust funds and benefit                programs could not tolerate another interruption of claims-taking                activities.  Based on the experience during the November lapse                in appropriations and the loss of four full days of production time,                any further interruption in service would have a devastating long-term                impact on SSA’s ability to process Social Security, SSI and Black                Lung claims.  The Agency was still attempting to recover from                the effects of the November furlough.  Therefore, employees                in direct service positions would remain operational, while staff                support employees would be furloughed.  When the partial shutdown                began December 16, 1995, about 55,000 Agency employees, most of                whom processed claims and/or provided direct public service, were                told to report for work.  A total of approximately 11,000 staff                employees remained furloughed.