Serial port is not suitable for programming serial eeproms (unless eeprom has a uart interface). Commonly, serial eeproms (eg AT93C46) are programmed via 3-4 pins (which is not RxD, Txd):

DI:Data in
DO:Data out
SCK: Clock
CS : Chip select

Their timing charts and programming instructions are given in the datasheets. Various code security methods can also be implemented like CRC, LRC to avoid data loss or miss+(writing/reading).