No, do not take out the voice box as that does not fix the dogs actual problem with regards to separation anxiety and crying.
If the dog is in a crate, cover it. That tends to soothe the dog a bit. The dog will also recover the more it gains confidence and the realization you will always come home.
Which reminds me: when you leave and/or come home, do not make a big deal out of it. Simple put the dog in the crate with no fussing when you are about to leave and when you come home, let the dog out of the crate with no hello's, petting, etc. Pretty much ignore them for the most part.
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