The famous GhantaBro jokes apart (and with his ever so often reference to meet hot girls...one hopes his wife would really not read this forum ), answer to quietmouse's question is "You are not required, but good idea to let the HR department know so they can update the I-9 form"
The employer's instructions for the I-9 (found in M-274.pdf: http://www.uscis.gov/files/nativedocuments/m-274.pdf) says the employers should not reverify green cards or US passports that have already been presented with an I-9. So you don't have to update the employer if you have already filled out an I-9 using your green card, unless there are positions in the organization that you are interested in which are only available to US citizens.
By law, no, you do not have any such obligation. I was friends with my boss (now my ex-boss) and on the day of my oath he gave me the day off without taking it off my annual entitlement.
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