Support and Defend Our Military Personnel and Their Families Act This bill states that any person who serves or 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. The Immigration and Nationality Act (INA) is amended to extend the period for filing a naturalization application to one year after completion of eligible military service. An alien who is eligible for a family-sponsored immigrant visa and is either the spouse or child of a permanent resident alien who is serving in the Armed Forces shall be exempt from worldwide immigrant 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 is serving or has served in the Armed Forces under honorable conditions. The bill permits posthumous benefits under specified circumstances. With respect to a removal proceeding under INA: a notice to appear shall not be issued against an alien who serves or has served under honorable conditions in the Armed Forces without prior DHS approval; DHS, in determining whether to issue a notice, shall consider the alien's eligibility for naturalization, military service record, grounds of deportability, and any hardship to the Armed Forces, the alien, and his or her family if the alien were to be placed in removal proceedings; and an alien who serves or has served under honorable conditions in the Armed Forces shall not be removed from the United States on specified grounds.