Most of the pics of models I've seen have round bases - is the game base-specific, or would models on square bases do as well?
And is the base size relevant?

Generally round bases are a lot better for skirmish games (where the models are not in formation).
If you have square bases and melee is defined by base contact you have to define what 'base contact' is, and it becomes trickier to judge. As is measuring.
With round bases there are no such difficulties.
Base sizes tend to be more forgivin, as long as all are the same size. Combining different sizes means that some are better at outnumbering (and harder to outnumber), and moving in terrain.
In friendly games these things should not really be a problem.