Status: Half right.
First spotted: Twitter quote engine.
Shakespeare certainly said “Nothing will come of nothing.” It’s from King Lear. The old king is dividing up his kingdom among his three daughters, and has asked that they all compete to see who loves him more. Cordelia, the youngest, loves her father the most – but does not know how to explain that. So when Lear says “What have you got to say?” she replies, “Nothing.”
“Nothing will come of nothing,” says the king, which in this context means “I won’t give you any of the kingdom at all if you don’t come up with something nice to say to me, so try it again.”
When used by quote engines they like to treat it like it means, “You won’t ever get anywhere in life if you don’t try new things.” Hence the second and incorrect part of the quote that gets sent around, “Dare for mighty things.” From my research it appears that this portion of the quote is actually from Teddy Roosevelt:
“Far better it is to dare mighty things, to win glorious triumphs even though checkered by failure, than to rank with those timid spirits who neither enjoy nor suffer much because they live in the gray twilight that knows neither victory nor defeat.”