Okay, these are all the birthdays I could find for the Hetalia characters. If you know any more, or find any errors, please comment with the countries' name and their birthday.
Monaco: January 8th
Prussia: January 18th
Netherlands: January 30th
Greece: February 3rd
Japan: February 11th
Spain: February 12th
Lithuania: February 16th
Estonia: February 24th
Egypt: February 28th
Australia: March 3rd
North Italy: March 17th
South Italy: March 17th
Belgium: April 19th
England: April 23rd
Seborga: April 23rd
Norway: May 17th
Cuba: May 20th
Ladonia: June 2nd
Denmark: June 5th
Sweden: June 6th
Hungary: June 8th
Iceland: June 17th