I personally believe this rumor is false, for the following reasons.
[*]
It's illegal in the US/UK.
If you are found to be using a flash cart or third-party accessory, Nintendo has the right to stop you from using their online services, because that is a service. However they do not have the right to brick your device, as it is a good. The law recognizes the clear difference between goods and services. While some may argue that the software on the device is still Nintendo's property that you're licensing, there's two counters.
- The hardware will not function as advertised without the software, so if Nintendo did purposely cripple the software or brick the device they could get sued in a heartbeat. For an easy example, check out the lawsuits Sony is under for removing a major function of their PS3 system software. This can be found under US court documents 3:10-cv-01975-RS and 3:10-cv-01811-EMC, though later some of the suits merged. In the UK people could skip the suit and were able to request a partial refund for their machine once OtherOS was removed, thanks to UK's consumer protection laws.
- If Nintendo has the right to do this because you're just licensing the OS of the machine, then Microsoft has the right to brick your computer if you install something they don't like. Obviously this is not true and would never fly in a court of law.
[*]
Other systems also warn against third-party modifications.
Checking out Nintendo's manuals (the PDFs for recent systems are available on their site), you'll notice that the GBA SP, GBA Micro, Gamecube, DS, DS Lite, and DSi/XL all warn against unauthorized third-party modifications.
You'll also note that the warning gets more severe as time goes on, up to where the DSi/XL actually has three warnings on one page, and even says that if you mod it people could collect your personal information! This just SCREAMS "fear mongering".
[*]
The 3DS is already out in Japan and no such brickings have occurred, even though some of the most popular flash carts are being used on Japanese 3DS systems.
Don't even need an explanation here...