Where does the Squire fit in?

For a detailed look at each of these characters, simply follow the link provided below to access Gradesaver's summary of the prologue. In this summary, you will find the order (from highest to lowest) or status of each of the pilgrims. I assume that this is what your question refers to as far as "fitting in." Chaucer's pilgrims represent the different rungs of English society. The summary will supply you with this information in order of importance. 

The Knight brings with him his son, The Squire, a lover and a lusty bachelor, only twenty years old. The Squire cuts a rather effeminate figure, his clothes embroidered with red and white flowers, and he is constantly singing or playing the flute. He is the only pilgrim (other than, of course, Chaucer himself) who explicitly has literary ambitions: he 'koude songes make and wel endite' (line 95).