I don't know about head and neck cancer, but there is a lot of work going on trying to figure out why some cultures have very little breast or prostate cancer, and why others have so much. An epidemiologist at Hopkins once called these and some others, "lifetstyle" cancers, which includes diet (although genetics can also play a role). As regards diet, cultures where little red meat is eaten, compared to lots of vegetables, healhty oils like olive oil, fish and soy, seem to have the least problems. There have been some studies relating increased growth of prostate cancer to certain animal fats, for example. What is interesting is that immigrants from countries where prostate cancer is (or was) relatively rare, their offspring -- eating a American diet -- end up with same rate of PC as the rest of the US population (where it is the most common cancer for men, after skin cancer).
For this reason a Mediterranean-type diet or oriental diet, with a lot of healthy plant oils, soy, vegetables and fish or poultry rather than red meat, is often recommended to men who are at risk for PC, who have had it treated, or who are monitoring very low volume (so-called "latent" PC) with watchful waiting.
Whether this would have any benefit for head/neck cancer -- well, as Gary says, it makes you feel as if you are being proactive. The other benefits would be for blood cholesterol, heart health and weight.
Certainly eating foods known to contain antioxidants, as well as those associated with immune system functioning (e.g. Selenium-containing foods such as nuts) can't hurt. Trying mega-doses of supplements, on the other hand, is apt to be counterproductive and even harmful.
Gail