Hmmmm...
I think from a formality standpoint it would be pretty much ok, but on a practicality standpoint, I wouldn't.
Basically, scrubs, as a professional uniform, add a certain amount of professionality (is that even a word?) and dignity to your look, which is complemented by our badge. On the other hand, the fact that you are even a little bit in danger of getting messy/dirty would make me inclined not to wear it.
Maybe you could compromise and wear a crest pin or your badge on a pendant? Or possibly one of the small badge-replica pins, which are appropriate on some items of clothing that the badge really isn't.
On a completely frivolous note, they're called scrubs, and therefore must be scrubby, and therefore not for badges!
