It's an objectively silly quote, but it's primarily (I would guess) used to motivate kids. For its purpose it's probably pretty good. Used as some kind of wisdom it's laughable.
"Winners never quit and quitters never win"

As I said, they're not completely bad, they have their purpose.

I am someone who plays on if there is any chance of saving the game. There does come a point when, in my opinion, it makes no sense to continue. Where that point is depends upon many things. If you have given away a rook to a GM I think it is time to throw the towel in. But if you lose the exchange against some rated a few hundred points below you, then why not continue? You may get back in the game. Of course, it all depends on the position too.
I do understand that some people get frustrated when people play on in hopeless positions - though often the people getting frustrated are quite weak players where there ego is punctured by their opponent playing on ('it is an insult that they think I could lose from here!!"). Secretly, they think they may blow it!!
But certainly people do play on sometimes way beyond the point where it is sensible. But I find just as many people resign in positions where there is plenty of play left. At the end of the day, live and let live - let people do what they want within the rules of the game.

I almost said this phrase to some children at chess camp this afternoon, but then remembered that I'm not Vince Lombardi and they are not football players.

http://www.positivityblog.com/index.php/2009/02/17/why-knowing-when-to-quit-can-lead-you-to-success/ Why knowing when to quit can lead to success
"If you learn from losing it isn't a loss." -BG Jett