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1.
The fighting area for participants in a combat sport. Origianally, these sports took place in a circle. When support ropes were added to clearer identify the border, the rings were made sqaure or rectangular in shape.

See also ring, boxing ring, wrestling ring, enter the ring, queensbury
As these two competitors enter the sqaured circle, they begin to realise that for one of them, fame and fortune on a tremedous scale awaits after this bout.
by kung-fu jesus August 19, 2004
 
2.
A U.S. Marine term for getting stuck on a Naval Ship that patrols one area for far too long of a time.

When a Naval ship gets sent to an area they are given a grid or "Square" that they are allowed to move around in or patrol. They do not leave this area unless assigned to move on. So when they get assigned to one grid for a very long time they are in essence just doing a bunch of circles patrolling that zone.

Hence the term: Squared Circles
"We got stuck doing squared circles in the Gulf for 3 months until we were finally able to hit a port!"
by purenrgy December 29, 2009