1999 Ford F-250 Rear differential failure

About a month ago I had a flat tire on the rear axle of my 1999 Ford F-250 XLT. I used the full size spare which was the same size as other tires currently on the truck. 2 days later the differential started making noise and had to be replaced. I asked at the dealership if they could tell me why it failed. They told me that it was because I used my spare tire on one side of the rear axle. Because the spare had no wear and the other tire was worn, and because the tires were made by different manufacturers in different years that the tires circumferance was different, and that driving for 2 days like this caused the problem. I've had friends tell me this is not true, since race cars use stagger and this is done by running different size tires, and that the purpose of the differential is to compensate for this. I am now going to buy new tires and need to know who's right so I know whether to buy 4 or 5 tires.