I use Remote Desktop on my home PC to connect to two different PCs at the office. On one of the remote PCs, my extra mouse buttons work, and on the other remote PC, the extra buttons sadly do nothing. All three PCs use X-Mouse Button Control to map the extra buttons to Copy and Paste. The home PC has Windows 7 Home Premium, and both office PCs have Windows 7 Professional.
I'm dubious about the answer on this post because I don't have a USB tunnel on either PC. It seems like a difference in configuration between the two PCs.