Like mdewey commented, "Polk" and "Pierce" are proper nouns referring to, respectively, James K. Polk and Franklin Pierce.
Polk was the Democratic Party nominee in the 1844 Presidential election, which he won. Pierce was the Democratic Party nominee in the 1852 Presidential election.
Separately, "poke" and "pierce" are both verbs that convey the image of something prodding or sticking something else, which can be used negatively if some person is the one being poked (or pierced). "Polk" is pronounced similarly to "poke" and "Pierce" is, of course, pronounced the same as "pierce."
So "We Polked you in '44; we shall Pierce you in '52!" is a campaign slogan in the form of a pun playing on the fact that Polk won the 1844 election and (so the slogan claims) Pierce will resoundingly win the 1852 election. The "you" could refer to the party's political opponents or to the country as a whole.