If you are following the "natural law" arguments which are the foundational basis for the birther objections, the question is as nonsensical as asking if congress had the power to redefine pi.
It is like suggesting congress has the power to define "arms" or "speech."
Congress has the power of "naturalization" which means "to make like natural." This is good enough for almost all intents and purposes, but it isn't the same thing. It is the distinction between an adoption and being actually born to a family.
The "natural law" principle argues that this characteristic is inherent, like eye or hair color, and can't be changed by appeal to earthly authorities. It is beyond their power to tamper with nature. They can only tamper with man made laws.
It is like suggesting congress has the power to define "arms" or "speech."
If the Constitution granted Congress the power to establish rules for keeping and bearing arms or speaking freely, I might ask those questions, but it doesn't. The Constitution explicitly prohibits Congress from establishing rules that infringe upon those rights.
Whereas, the Constitution grants Congress the power to establish a uniform rule of naturalization, which means not only that Congress has the power to say who must naturalize but also that Congress has the power to say who does not need to naturalize.
As Justice Gray wrote, there has never been any doubt that children born in a country of citizen parents are natural-born citizens. There have, however, been doubts as to all others. Thus the need for a uniform rule of naturalization.