Support and Defend Our Military Personnel and Their Families Act This bill establishes that any person who has served under honorable conditions as a member of the Armed Forces in support of contingency operations shall be eligible for naturalization as if the person had served during a period of presidentially-designated military hostilities. (Contingency operations are operations where Armed Forces members are or may become involved in military actions, operations, or hostilities, or that result in the call to active duty) The bill extends the period for filing a naturalization application from six months to one year after completing eligible military service. An alien eligible for a family-sponsored visa and is either the spouse or child of a permanent resident alien serving in the Armed Forces shall be exempt from worldwide visa numerical limitations. The Department of Homeland Security (DHS) may adjust to permanent resident status an alien who is a parent, spouse, child, son or daughter, or minor sibling of a person who has served in the Armed Forces under honorable conditions. The bill permits posthumous benefits under specified circumstances. DHS must give prior approval before a notice to appear in a removal proceeding may be issued against an alien who has served honorably in the Armed Forces. Before giving such approval, DHS shall consider factors such as the alien's record of service, grounds of deportability applicable to the alien, and any hardship to the Armed Forces or the alien or family members if placed in removal proceedings. Such an alien may shall not be removed based on certain grounds of inadmissibility or detained after ordered removed.