I also think the postal employee is confused. There are specific regulations for both live poultry and eggs:
For live poultry, the regulations MANDATE express mail:
http://pe.usps.com/text/pub52/pub52c5_007.htm#ep197790
If you will note, for the other categories of live animals the USPS will ship, the regs will also specify the method of mailing acceptable (priority, express, parcel post, etc).
For eggs, it really doesn't specify a method. Here is the ENTIRE regulation about shipping eggs. Since no method is mandated, I would interpret that to mean that any method of the shipper's choice should be acceptable, express, priority, first class and/or parcel, subject to the normal restrictions of weight and package size, of course:
54 Eggs
Eggs are mailable in domestic mail subject to the following general conditions:
1. They are individually cushioned.
2. They are otherwise packed to withstand shocks encountered during normal Postal Service handling.
3. They are not likely to be harmed by anticipated temperature changes while in Postal Service custody.
This is all in USPS Publication 52 - Hazardous, Restricted, and Perishable Mail.
If the postal employee still gives you a hassle, tell them to check Publication 52. If that still doesn't work, I'd ask to speak to the postmaster or supervisor on duty and see if that works. Or, worst case scenario, try shipping from a different post office if practical.
http://pe.usps.com/text/pub52/pub52c5_009.htm