This claim was made popular by being said in the movie The Social Network. It exactly says:
Did you know there are more people with genius IQs living in China than there are people of any kind living in the United States?
Skeptics Stack Exchange is a question and answer site for scientific skepticism. It only takes a minute to sign up.Sign up to join this community
This claim can actually be tested very easily. Others have done it, e.g. Brian Dickerson: The perils of repeating political nonsense and came to the conclusion that the claim was wrong.
You just need the chance a random person from a large group has a genius level IQ, then multiply it with the number of people in China and compare the result with the number of people in the USA. The IQ follows a Gaussian distribution where 100 is the mean by definition and a standard deviation (SD) of 15 (after Wechsler) or 16 (after Stanford-Binet). (sources: , ). The rarity of a particular IQ can be calculated easily and are listed by . This numbers give the percentage of people with an IQ equal or lower than the given IQ and the rarity of this occurring.
The main question actually is what an genius level IQ is. According to  it is 140 or higher ("Genius or near genius") while  states it as at least 160. The rarity of such an IQ is 1/261 (with a SD of 15) or 1/161 (SD=16) for an IQ of 140 and 1/31,560 (SD=15) or 1/11,307 (SD=16) for an IQ of 160 .
The population of USA for 2012 is about 313,221,000 according to the U.S. Census Bureau . The population of China for 2011 is about 1,370,537,000 . Taking the lowest rarity of 1/161 from above then there are about 1,370,537,000/161 = about 8,513,000 geniuses in China, which is significantly less then the 313,221,000 Americans. If you take the 15 SD value of 1/261 then you get only about 5,251,000 Chinese geniuses. With the 160 IQ points, which are more realistic  to indicate a real genius, the numbers are even smaller (about 43,400 or 12,100). Note that 1,370,537,000/313,221,000 = 4.375, so in order for this claim to be true actually one of every four (or say five) Chinese would need to be a Genius. Even without the numbers above, it can be easily understood that this is not the case.
Conclusion: This claim is wrong according to the definition and probabilities of the IQ.
It has been pointed out, that there are published research results  showing that China and other Asian countries have a higher average IQ of about 106 (or even as high as 115) as the normed average of 100. This would increase the number of geniuses in China: The above factor of 1/161 would drop to about 1/60 or, for a Chinese average of 115, even to about 1/17. These numbers can be easily calculated using the standard deviation, e.g. using MS Excel's function
NORM.DIST(140, 106, 16, TRUE). However, both these factors are not in any way close to the US/Chinese population ratio of about 1/4.375, so the original conclusion still holds.
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?