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Your own link (from the summary) states that it's a mixed bag at best:

> The relationship between obesity and socioeconomic status differs by sex and race and ethnicity group. Among women, and specifically non-Hispanic white women, obesity prevalence increases as income (PIR) decreases, while among non-Hispanic black and Mexican-American men obesity prevalence decreases as income (PIR) decreases. Although the prevalence of obesity among women with income below 130% of the poverty level is higher than among those with higher incomes, most obese women do not have incomes below 130% of the poverty level.

And considering education tends to (roughly) map to income, the most damning sentence is the last:

> Among men and women with a college degree, the prevalence of obesity is lower than among those with some college education. Moreover, college educated women are less likely to be obese compared with those with less than a high school diploma.

While OP may be incorrect in claiming most poor Americans are overweight, it doesn't look like they're too far off claiming those in poverty tend to be heavier. Doubly so considering this evidence is pulled solely from the link you're trying to use to refute their claim.