(Note: I don't know who Ann Coulter is. I can, however, comment on the generic claim that radiation is "good for you".)
There is empirical evidence that suggests that low to medium amounts of absorbed gamma radiation boosts immunity and resilience to ailments such as heart disease, though it may (or may not) increase rates of cancer. It has been suggested and that the reduction in the probability of death from other diseases offsets the increased probability of death from cancer.
The reference I found on this is Lawrence Solomon's opinion piece in the Financial Times: Low exposure to the Nagasaki atomic blast resulted in longer lifespans, which states:
The tens of thousands more distant from Ground Zero [of Nagasaki and Hiroshima], and who received lower exposures to radiation, did not die in droves. To the contrary, and surprisingly, they outlived their counterparts in the general population who received no exposure to radiation from the blasts.
The only evidence that exists as to the health of humans who have been irradiated at low levels points to a benefit, not a harm. Difficult though it may be to overcome the fear of radiation that has been drubbed into us since childhood, there is no scientific proof whatsoever to view the radiation emitted from the Fukushima plant as dangerous to the Japanese population
These findings are supported by (and reference) the paper "Mortality of A-bomb Survivors in Nagasaki and Hiroshima", by M. MINE, S. HONDA, Y. OKUMURA, H. KONDO, K. YOKOTA and M. TOMONAGA, Atomic Bomb Disease Institute, Nagasaki Univ., Sch. Med., Nagasaki 852-8523, JAPAN. This paper states:
From the analysis of LSS’s population, RERF, lower relative risk of mortality from non-cancerous disease than control was observed for a dose range, 0.06-0.49Gy, when city was not adjusted. But when city was adjusted, lower relative risk was not observed. Although the number of subjects analyzed in Nagasaki University was smaller than that of LSS’s population, we have obtained the lower relative risk of mortality from noncancerous disease for male at a range of low doses.
These correlate with my undergraduate studies in physics and later work as a health and safety worker for Atomic Energy of Canada Limited (AECL). In undergrad I recall hearing that Marie Curie and her husband wore on their arms slabs of radium and/or uranium because they were under the impression that the radiation these materials emitted boosted immunity. As a worker for AECL at a nuclear power plant, it was commonly accepted that people working at the nuclear power plant on average lived longer than the general population and had lower rates of cancer (Submission of Dr. Patrick Moore, Chairman & Chief Scientist Greenspirit Strategies Ltd. To The U.S. Nuclear Regulatory Commission Public Hearing on Potential Environmental Impacts From a License Renewal of Indian Point Nuclear Power Plant, dated September 19, 2007). From that reference:
A 2004 Columbia University Study of 35,000 respondents concluded that “…nuclear power plant workers in the United States…live longer and have significantly lower cancer rates compared to the general population.”
While it doesn't seem to be the study being referred to above (as the sample sizes are different), there is a 2004 study from Columbia University: "Analysis of the Mortality Experience amongst U.S. Nuclear Power Industry Workers after Chronic Low-Dose Exposure to Ionizing Radiation", which states:
The cohort displays a very substantial healthy worker effect, i.e. considerably lower cancer and noncancer mortality than the general population. Based on 26 and 368 deaths, respectively, positive though statistically nonsigniﬁcant associations were seen
for mortality from leukemia (excluding chronic lymphocytic
leukemia) and all solid cancers combined, with excess relative
risks per sievert of 5.67 [95% conﬁdence interval (CI)22.56,
30.4] and 0.506 (95% CI 22.01, 4.64), respectively. These estimates are very similar to those from the atomic bomb survivors study, though the wide conﬁdence intervals are also
consistent with lower or higher risk estimates. A strong positive and statistically signiﬁcant association between radiation
dose and deaths from arteriosclerotic heart disease including
coronary heart disease was also observed in the cohort, with
an ERR of 8.78 (95% CI 2.10, 20.0).
Whether this is the study referred to or not, it supports the conclusion asserted in the paper.
All that being said, there are significant deleterious effects from significant radiation exposure. For example, while the Curies strapped radioactive materials to their arms on the basis that it benefitted their immunity, they suffered significantly for it (Marie Curie: Pioneering Physicist By Elizabeth R. Cregan, p. 17).
While gamma radiation is, on the above basis, understood to have some benefits, the effects of beta and alpha radiation as are known to be quite destructive to cell tissue. While these forms of radiation are less common to be exposed to and easier to protect against than gamma radiation, I strongly doubt they've any positive effect at reducing the risk of cancer. The one known exception may be exposure to Iodine-131, a beta particle emitter, because it collects in the thyroid. While high doses will destroy the thyroid and stop iodine from collecting in the thyroid, low doses will collect there and the beta particles released by the iodine will cause thyroid cancer.
Going back to the claim that those around Fukushima are "much less likely to get cancer" seems overstated. While evidence supports the existence of certain health benefits, and the evidence is mixed about whether exposure to radiation increases or decreases risk of cancer, I feel it's reasonably clear from the evidence we have at present is that the risk of cancer wouldn't be "much less likely", as claimed.
As well, on whether cells have a natural defence mechanism when exposed to low or medium doses of radiation: I don't know of any studies, nor do I have any other knowledge or information, that would support such a conclusion. That being said, I think it's perfectly plausible that cells could have such an ability.
I hope that's helpful.