No, rotten eggs won't make you live longer.  I would never recommend eating one, either.  It's just the smell of rotten eggs, the gas hydrogen sulfide, that may provide you with more years to live.  How's that?  Hydrogen sulfide may activate a gene inside your body's cells that are connected to longevity.

Hydrogen sulfide was used for a short period of time during World War I as a chemical weapon because it's toxic and has explosive properties.  So how in the world can it prolong life?

If it's used in proper doses, data suggests that if may be able to prevent the symptoms of aging and diseases associated to aging by stimulating klotho, a gene that is thought to promote antioxidants within the body.  Unfortunately there is also a link between hydrogen sulfide and high blood pressure, Parkinson's disease and Alzheimer's disease.  So, obviously, the key is to get Klotho "turned on" without promoting the negative affects of hydrogen sulfide.  Investigation into the use of hydrogen sulfide continues with the hope that this can be accomplished.

I don't recommend you stock up on rotten eggs to sniff every day since that won't get you anything but a stinky house.  Then again, you're dog may like it!