This phase I trial studies the side effects and the best dose of viral therapy in treating patients with squamous cell carcinoma of the head and neck that has returned (come back) after a period of improvement or has spread to other parts of the body or breast cancer that has spread to other parts of the body. A virus called encoding thyroidal sodium iodide symporter, which has been changed in a certain way, may be able to kill tumor cells without damaging normal cells.

PRIMARY OBJECTIVES: I. To determine the maximally tolerated dose (MTD) of intratumoral administration of an Edmonston strain measles virus genetically engineered to express human thyroidal sodium-iodide symporter (NIS) (oncolytic measles virus encoding thyroidal sodium iodide symporter [MV-NIS]) in patients with recurrent/metastatic squamous cell head and neck cancer. II. To determine the safety and toxicity of intratumoral administration of MV-NIS in patients with recurrent/metastatic squamous cell head and neck cancer and metastatic breast cancer. SECONDARY OBJECTIVES: I. To assess in a preliminary fashion antitumor efficacy of this approach by following, radiographic response, and time to progression. TERTIARY OBJECTIVES: I. To determine the time course of viral gene expression and virus elimination and biodistribution of virally infected cells at various time points after infection with MV-NIS using single-photon emission computed tomography (SPECT)/computed tomography (CT) imaging. II. To assess viremia, viral replication, and measles virus shedding/persistence following intratumoral administration. III. To determine humoral and cellular immune response to the injected virus. OUTLINE: This is a dose-escalation study. Patients receive oncolytic measles virus encoding thyroidal sodium iodide symporter intratumorally (IT) on day 1. After completion of study treatment, patients are followed up at 6 weeks, every 3 months for 1 year, and then every 6 months for 1 year.

Inclusion Criteria:

• Pathologically confirmed squamous cell carcinoma of the head and neck OR pathologically confirmed invasive breast adenocarcinoma with documented estrogen receptor (ER)/progesterone receptor (PR)/human epidermal growth factor receptor 2 (HER2) status and radiographic evidence of distant metastatic disease
• Measurable disease
• Head and neck cancer OR metastatic breast for which standard therapy is not curative *NOTE: Patients with ER/PR positive, HER2 negative breast cancer must have progressed through at least one prior cytotoxic regimen for advanced disease and no longer be candidates for standard endocrine therapy; patients with HER2 positive breast cancer irrespective of ER/PR status must have received or no longer be candidates for standard HER2 directed therapy (i.e., trastuzumab, pertuzumab, trastuzumab emtansine, and lapatinib); patients with ER/PR/HER2 negative breast cancer must have progressed through at least one prior cytotoxic regimen for advanced disease; ER/PR and HER2 status are defined by current American Society of Clinical Oncology (ASCO)/College of American Pathologists (CAP) guidelines
• Patient may have more than one site of recurrence/metastatic disease but only one lesion will be injected that is >= 1 cm in size (if in the lung, the lesion must be >= 2 cm and adjacent to the pleura in the lung)
• Absolute neutrophil count (ANC) >= 1500
• Platelet (PLT) >= 100,000
• Hemoglobin (HgB) > 9.0 g/dL
• Total bilirubin =< institutional upper limit of normal (ULN)
• Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase [AST]) =< 1.5 x ULN
• Creatinine =< 1.0 mg/dL
• Negative pregnancy test done =< 7 days prior to registration, for women of childbearing potential only
• Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
• Provide informed written consent
• Willingness to return to Mayo Clinic enrolling institution for follow-up
• Willingness to provide biologic samples for correlative research purposes
• Life expectancy >= 12 weeks

Exclusion Criteria:

• Any of the following * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception during treatment and 8 weeks following the completion of treatment
• Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
• Receiving therapeutic anticoagulation (Coumadin or low molecular weight heparin)
• Active infection =< 5 days prior to registration
• History of tuberculosis or history of tuberculin purified protein derivative (PPD) positivity
• Any of the following prior therapies: * Chemotherapy =< 3 weeks prior to registration * Immunotherapy =< 4 weeks prior to registration * Biologic therapy =< 4 weeks prior to registration * Radiation therapy =< 3 weeks prior to registration
• Failure to fully recover from acute, reversible effects defined as =< grade 1 Common Terminology Criteria for Adverse Events (CTCAE) version (v.) 4.0 of prior chemotherapy regardless of interval since last treatment
• Requiring blood product support
• Central nervous system (CNS) metastases or seizure disorder
• Human immunodeficiency virus (HIV)-positive test result, or history of other immunodeficiency
• History of organ transplantation
• History of chronic hepatitis B or C
• Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration [FDA] approved indication and in the context of a research investigation)
• Treatment with oral/systemic corticosteroids, with the exception of topical or inhaled steroids
• Current exposure to household contacts =< 15 months old or household contact with known immunodeficiency
• Willing to avoid household contacts =< 15 months old or household contact with known immunodeficiency 1 week after treatment
• Allergy to measles vaccine or history of severe reaction to prior measles vaccination
• Allergy to iodine; Note: this does not include reactions to intravenous contrast materials