My first contact was with a recruiter, who was a bit disorganized - he missed our first phone call due to meeting schedules, a sign of how busy things are at Square. After some chitchat, he scheduled a Skype interview with an engineer.
The Skype interview was fun, actually. Square has an Evernote-like online wordpad, on which the interviewer and I wrote code. The task was to implement a function from the C standard library, going through several iterations. The interviewer was also a bit disorganized, but was thoroughly pleasant and engaged throughout the hour-long process.
The interview went well, and within a few hours the recruiter contacted me to set up an in-person battery of interviews for the next week.
The in-person interviews took place at the main office in San Francisco, which is thoroughly modern and attractive. A series of 6 different interviews ensued, with a roughly equal mix of 'discussion' (read: whiteboarding) interviews with pairs of Square employees and pair-programming with single interviewers. The discussions ranged from in-depth discussions of system architecture to sketching out solutions to given problems. There was a pleasant lack of 'academic' challenges in the discussion phases.
The pair programming exercises ranged from academic to practical, but the experience was good. The engineers were pleasant, and I never felt pressured or judged (despite the fact that an interview is, by definition, an exercise in judgement).
After the formal interviews, I was told immediately that an offer would be made. I was introduced to some engineers and had dinner with one. A formal offer was made the following morning.
On the whole, the hiring process was an odd combination of relaxed and rapid. I liked everyone that I met, had interesting discussions and challenges, and the whole process took 12 days - a record in my experience.