I take issue with these kind of quotes because, to be quite frank, they are completely out of context and make no sense in the myriad of ways people try to make them apply to unrelated contexts. It sounds like nothing more than marketing hype, as opposed to a message of substance.
The point isn't that only full-timers can be founders, but rather that all founders should share a roughly equal level of commitment. If some of the founders are full-time, and some are part-time, then you've got a really severe imbalance in terms of contributions to the company, not to mention risk.
This will cause huge bitterness later on, so it's totally reasonable to say "We're all going in full-time, so if you want to be a founder, you are too. If not, that's fine, and we'll make you hire number one when we can afford to pay you a salary, plus back-pay for hours worked as an IOU."
The quote is, of course, over the top, after all we're not fighting for our lives in a foreign land. But I don't think it's that out of context: I'm 42, have a child, pay a high rent and support my larger family. For me to quit my daily high-paying job to pursue what mat turn out to be a dream does require a huge amount of courage. If the company doesn't succeed, say, after 2-3 years, I may not be able to find another job like this. That's why quotes like that resonate with me.
Please understand that not all would-be founders are 25 year-olds with little to lose.
Personally, I believe that having a day job while working on a startup should not be held against you. If everyone agrees that you're a co-founder, then you deserve co-founder equity.
Do you have less work to do because you have a day job? No, not necessarily. Were you there from the beginning? Yes. Are you involved in founder-like meetings and decisions? If yes, that should be qualification enough.
It was scary, but we did specifically did not burn our bridges. We kept in touch with all the people we could so that if we needed to have some bridge funding via consulting work, it was always on the table for us.
The strongest counter-argument I can make for saying "You need to be doing X,Y,Z to succeed" is that the failure rate for startups is over 90%, so no-one knows what really works or doesn't. There's a combination of execution, luck, and recognizing good timing that comes into play. I've seen companies/individuals hit it big with the most laid back attitude toward the product because they found the right thing at the right time (PlentyOfFish), and people who work insanely hard who just disappear (the list is huge here...).