The construction of (associative) square extension of an idem-potent groupoid (semigroup) was recently introduced by A. W. Marczak and J. Płonka. They proved that all square extensions of a variety of idempotent groupoids also form a variety. In this note we explicitely describe the free objects in semigroup varieties obtained from band varieties by means of associative square extensions, which is done by solving the corresponding word problems. As a consequence, we calculate the free spectra and the $p_n$-sequences of the considered varieties.