1. To confront someone in defiance or hostility or in preparation for a fight. He squared up with against guy who had been making lewd remarks about his girlfriend. Protestors squared up against police during the march, but thankfully there was no violence.

2. To face someone in competition. The heavyweight champion is squaring up against his challenger this Saturday at 2 PM. The last time I squared up against her, she only beat me by a couple of points. This time, I'll come out on top!