We send fresh faunal materials for DNA typing, etc. overnight mail.  We
package them in a ziplock bag with a wet papertowel to keep them moist and
enclose them in a crush-proof box.  So far, they have all arrived as
expected.
 
Botanical Specimens might be shipped UPS if re-enclosed in a flower press for
stability.  With plenty of blotters and ventilators, environmental conditions
should remain very stable.
 
For extraordinarily fragile specimens, I would ship in a styrofoam container
the fastest method possible.
 
Good luck!