Do unique and square have a long U vowel sound?

No, only "unique" has the long U at the start. (yoo-neek)

In unique, the U has a long U sound, the I has a lone E sound. (QUE has a K sound)

The U in "square" pairs with the Q to make a KW sound, then the A is a caret A (long A + R = air). The E is silent.