The answer is no. I've played early and late prototypes from dozens of published boardgame designers including several who are successful enough to do it for a living and I've never been asked to sign an NDA by any of them. Boardgame ideas just aren't worth stealing.
The only time I've been asked to sign boardgame NDAs is by unpublished designers who were inordinately fond of their ideas (no insult intended). Most ideas don't actually turn into publishable games, and even when they do, a very small percentage of them actually get published.
Worse, boardgame ideas aren't actually protectable: the only law that protects ideas is a patent, and no boardgame patent has been successfully defended in court (the only exceptions are games with mechanical designs, like Hungry, Hungry Hippos or MouseTrap). If you want to see money wasted search the patent office for boardgame patents: I've paged through hundreds and have yet to find one that actually became a recognizable published game. I know of two examples of patented games that were published, and only one of them, Magic: The Gathering went on to make any money.
Don't worry about your idea being stolen. Focus on playing the game with as many people as possible, and then when you think you've got something publishable, get as many people as you can that you don't know to play it without you there, just with rules and a prototype (called "blind playtesting").
If your idea survives all of the changes you'll discover you need to make after you work through the results of blind playtesting, then start approaching publishers. You'll discover that none of them will sign NDAs either, though for a more practical reason: they may well have something in development that is astonishingly like your game, and they'd lose the ability to publish it due to the NDA. That, or they've already received submissions that are remarkably similar to yours, with the same problem.
It turns out that most ideas, including game ideas, build on what already exists, consciously or not, and that very similar or even identical ideas come to multiple people at about the same time, as they're exposed to much the same things. It doesn't mean it's not worth pursuing, it's just a matter of the reality of ideas.
Anyway, I don't mean to discourage you; rather, I encourage you to go forth and test, redesign, and test some more until you've polished something really great. And then I look forward to playing it!
Note: I am not an attorney, so do not take what I've told you as legal advice. I am relaying what I've learned from publishers and designers while involved in the game design world over the last ten years.
