Accepted into EPFL, might not get Visa in time, what am I signing myself up for?
I know of international students in my research lab (in Canada, not Switzerland) that crossed the US borders and came back immediately to renew student visas. In itself, this is not illegal. 

However, I don't think that you would be allowed to study in your program without a student visa, because those are usually required by the university to receive insurance coverage (for instance, if you're working in a lab). You really should ask your program director for advice on this situation. I would expect them to be understanding and guide you.