So the main question I have is while the serial data bytes are going into the buffer during the interrupt, the program can still run doing something else like program the bytes into the eeprom?
NO, but when the ISR end, the "program" will be resumed. There is something here, this can NOT be your question, elaborate. If it is, get some veru basic litterature and STUDY.

Interrupt driven serial routines involve much more overhead as far as program code size and RAM requirements
MUCH?????



In the time it takes to write a byte to the eeprom, if an interrupt occurs (like in between a pulse,

or when bits are shifted out) will it still process ok once the interrupt is complete?
depends on your code

If I use the polling method every 4rth line gets programmed in the eeprom, simply because the bytes are lost while the programming is executing.
That's whi interrupts were invented

Erik