I think Square and Stripe can easily co-exist - the more pressing issue, as far as I can see it, is Square's model won't scale very well
at all to EMV ('chip and PIN') cards, which severely limits their market.
Square's current magstripe reader is very simple, and thus cheap to produce to the point where they can give it away. EMV readers are not cheap. They must be certified, and are far more complex. I was in an Apple Store in London a few days ago and they were selling PayPal branded EMV readers for about $150.
Even if we assume a very healthy markup on that price, given the margins Square is on they cannot afford to give away a reader that would cost them ~$50 to produce. It changes their entire business model. Square have offered no indication of how they plan to support EMV cards once they become widespread in the US (which will eventually happen), other than 'you can still swipe the magstripe', which whilst true also usually means the fraud liability remains with the retailer, rather than the bank.