Hi,
It is hard to judge whether it would or not.
The process of the iMessages registrations seems to have taken on using this Serial Number which in turn leads to you getting a certificate (in the KeyChain) that allows iMessaging to work.
My initial thought was that this was to confirm the OS was on a Mac and not some install on a PC, "Hackintosh" style.
As the board should have a Mac/Apple Serial Number on it I would have thought it would have worked.
On the other hand there may be checks on used/released Serial Numbers and the engineer has not informed Apple that this one is in use.
An Apple Store or Apple Authorised Service Provider should be able to tell you.
Tomblah900 in this thread managed to get it done for free.
9:47 PM Tuesday; December 11, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously