For some reason each post office likes to make up their own rules on what to do with live chicks. Priority mail is fine. Here are the actual USPS rules on shipping live animals:
http://pe.usps.com/text/pub52/pub52c5_007.htm#ep184002
526.31 General
The following live, dayold animals are acceptable for mailing when properly packaged: chickens, ducks, emus, geese, guinea fowl, partridges, pheasants (only during April through August), quail, and turkeys. All other types of live, dayold poultry are nonmailable. Dayold poultry vaccinated with Newcastle disease (live virus) also is nonmailable.
526.32 Mailability Requirements
The specific types of dayold poultry named in 526.31 and DMM 601.9.3.2 are mailable subject to the following requirements:
1. Poultry that is not more than 24 hours old and is presented for mailing in the original, unopened hatchery box from the hatchery of origin.
2. Date and hour of hatching is noted on the box.
3. Box is properly ventilated, of proper construction and strength to bear safe transport in the mail, and is not stacked more than 10 units high.
4. Dayold poultry is mailed early enough in the week to avoid receipt at the office of address (in case of missed connections) on a Sunday, a national holiday, or the afternoon before a Sunday or national holiday.
5. Dayold poultry can be delivered to the addressee within 72 hours of the time of hatching.
6. If dayold poultry is sent at a Parcel Post rate, special handling postage must also be paid.
7. If dayold poultry is sent via air transportation (i.e., Express Mail, Priority Mail, or FirstClass Mail), all provisions of the airlines must be met. Delivery of the mailpiece is dependent on the availability of air carriers having available equipment to safely deliver the dayold poultry within the specified time limit.
8. Dayold poultry that is first shipped via a commercial air express or air cargo service and then presented for mailing to a final destination must be in good condition and properly packaged as specified in 526.32a-e.
9. Boxes of dayold poultry of about identical size, securely fastened together to prevent separation in transit, may be accepted for mailing as a single parcel, provided the total length and girth combined does not exceed Postal Service limits.