Two thoughts... first, Eaton used to recommend that synthetic gear fluid not be used. They have recently changed that, saying synthetic lubes are OK, and that the recommendations of the vehicle manufacturer be followed to protect bearings and seals. Here's their newest language, from page 24 of their installation manual - found at this link:
http://www.eaton.com/ecm/idcplg?IdcService=GET_FILE&allowInterrupt=1&RevisionSelectionMethod=LatestReleased&noSaveAs=0&Rendition=Primary&dDocName=PCT_1087652
Detroit Truetrac®
High quality mineral or synthetic gear lubes are required for use in Detroit Truetrac differentials. Regardless of the lube type, always use a GL5 rated lube with the least amount of friction modifier. Mineral lubes lacking friction modifiers (limited-slip additives) were historically recommended for all Truetrac applications because friction modifiers can slightly reduce the bias ratio (limited-slip aggressiveness) of Truetrac differentials. However, to address the continually increasing power outputs of modern powertrains, many vehicle manufacturers have switched to synthetic lubricants as a counter measure for increased axle temperatures and prolonged service intervals. In general, consult the vehicle owner's manual for the manufacturer's recommendations for lubrication type, weight and fill volume. This will ensure lube compatibility with the seal materials and bearings used in the axle. Eaton Performance technical support is available for any concerns in lube selection.
Second, I agree with BWV, the fluid isn't what would cause the whining. If the gears weren't set up appropriately, that would be the cause. Take it back to the shop and have them listen to it and address the issue. Sounds like its probably a reputable place. They should be able to handle it.
Good luck!