The claim is made that having regular influenza vaccinations helps one stay free of cancer. However, a search of PubMed reveals no such study to support the assertion.
Could influenza viruses cause cancer then? So, preventing their infection would then lower the risk for cancer? However, influenza viruses are not known amongst those known to contribute to cancer: 
About 15-20% of human cancers worldwide have viral etiology. Emerging
data clearly indicate that several human DNA and RNA viruses, such as
human papillomavirus, Epstein-Barr virus, Kaposi's sarcoma-associated
herpesvirus, hepatitis B virus, hepatitis C virus, and human T-cell
lymphotropic virus, contribute to cancer development
We do know that chronic inflammation can cause cancer, but as it says in this article 
Chronic inflammation may be a causative factor in a variety of
cancers. In general, the longer the inflammation persists, the higher
the risk of cancer. Hence, acute inflammation, such as occurs in
response to a transient infection, is not regarded as a risk factor
for the development of neoplasia, although many of the same molecular
mediators are generated in both acute and chronic inflammation. In
general, inflammatory leukocytes such as neutrophils, monocytes,
macrophages, and eosinophils provide the soluble factors that are
thought to mediate the development of inflammation-associated cancer,
although other cells, including the cancer cells themselves, also
and since the flu is clearly a transient infection, the claim is unsubstantiated.
 Poreba E, Broniarczyk JK, Gozdzicka-Jozefiak A. Epigenetic mechanisms in virus-induced tumorigenesis. Clin Epigenetics 2011 Aug;2(2):233-47. doi: 10.1007/s13148-011-0026-6. PubMed PMID: 22704339.
 Shacter et al. Chronic Inflammation and Cancer Review Article | January 31, 2002 | Colorectal Cancer, Oncology Journal