In nursing school, we were forbidden to wear our scrubs anywhere on the WAY to clinicals, as well as on the way home. I always came home, changed and showered immediately, and put the scrubs into their very own laundry bag. They were also washed separately from other clothes. There are some bad germs hanging out around hospitals, often in higher concentrations than out in the general public (that is why MRSA acquired out in the community became a news story).
Hospitals are germy, dirty places. I love microbiology and have a healthy respect for any and all microbes. I respect them enough to try to prevent being their vector for traveling out of hospitals (where they reside in larrrrrrrge numbers) into my home or into the community. It's the same reason I have the kids bring their own book to the pediatrician -- we do NOT touch the ones in the exam rooms/waiting rooms.
A virus absolutely CAN live on cloth (a flu generally longer than a cold virus, btw), so coughing into your sleeve doesn't mean the virus isn't on your clothing. It is, and can live there for a while. It's just that you don't touch light switches, doorknobs, etc with the inside of your elbow. A virus generally can live longer on hard surfaces, though.