Could Concorde have used swing wings?
Actually, the Boeing design for the SST (Super Sonic Transport), the Boeing 2707 had a swing wing (swing wing was the hot thing during that time). It is instructive to look at what happened to that design.



Image from up-ship.com

The 2707 concept was a larger aircraft, with a 174' wingspan for TO/landing and low speed regime (In comparison, Concorde had a 84' wingspan and carried half the passengers). While the wings were swept back, the wing area increased from 1828 $m^{2}$ to 2743 $m^{2}$.

The engines were carried on the stabilizers. During the development Boeing had significant problems associated with the swing wing mechanism. Concerns about stability and payload capability lead the design to be lengthened and added canard to the front to meet rotation requirements, though the new design 2707-200 kept the swing wings.



Image from up-ship.com

However, these changes did little to solve the problems. When kept near the centerline, the hinge mechanism interfered with the undercarriage. The hinge mechanism was so heavy that it negated the advantages of swing wing, leading to a very poor payload to weight ratio. Studies indicated that the aircraft would run out of fuel halfway across the Atlantic, essentially killing the project.

Due to this, Boeing decided to use a fixed wing platform in its next iteration, the 2707-300, which looked quite similar to Concorde.



"Boeing 2707-300 3-view" by Nubifer - Own work. Licensed under CC BY-SA 3.0 via Commons.