Why should I own a square?
Well first of all because it's a lot of fun. We don't know what will end up rendered on the board as a result of the collaboration of thousands of users and how it will morph into different images over time, but given the examples at What is it? we can imagine how it might shape up to be.
We'd like the board to be a free expression of the Internet. And while we are a centralized organization hosting a website, and have to follow the rules imposed and might need to censor what is displayed on InfinitumX, you'll be glad to know we've thought about this from the start of the project and have made the application itself fully decentralized. Technical details of which can be found here.
Additionally, squares can be profitable. Harberger's Tax makes sure the economy of the squares remains fair. Once you own a square and you pay your due taxes, then it's yours to use in any form you like. Maybe show your favorite meme in it, or a quote, a TV character, or... The logo of a famous fizzy drink for a cost you agree with them. Essentially you could rent it out as an advertising board.
More so, if the price that you've set is high enough and someone ends up snatching it from you, you will have made a profit in that sale. It's only if you choose to set a price lower than the price you set it for that you would make a loss in the sale itself.