Calvin the Ant refers to two of Calvin's alter egos.
Calvin the Ant as a human
Calvin the Ant appeared in a single Sunday strip as an ant-sized version of Calvin with insect-like behavior. He worked as a regular ant for an ant colony, presumably somewhere outside of his house.
Calvin the Ant as an antCalvin the Ant appeared as an actual (albeit anthropomorphic in behavior) ant in another Sunday strip. There, "sick of working all the time", he took a break from servitude and labor and lay down for others to do the work.
This alter ego was once again analogous to Calvin's reluctance to work around the house. His mention of 'the queen [ant]' refers to his mother.