Is there a way to get VirtualBox to work with an Intel 64 bit processor?

AMD64 is the x86_64 standard.

AMD licensed their 64-bit x86 implementation to Intel after Intel's non-backwards-compatible IA64 implementation was a market failure.

All Core 2 Duo and newer (all the i3, i5, i7) (exception: early Atom) Intel CPUs support AMD's AMD64 x86_64 implementation.

So just choose x86/amd64 (provided you are running a 64-bit OS)

Note: Often software referring as compatible with x86 is in reference to x86 32-bit. x86_64 CPUs can run either 32-bit or 64-bit OSes.