How to write an EPROM program
There is no such thing as an "EPROM program" ... you need to know what the EPROM supplies its data to. Any such information you can add to the question may help, even a good quality photo of the PCB. 

If that's a Z80 microprocessor (for example), there is such a thing as a Z80 program, which you may be able to disassemble (read the EPROM in the burner, write out as a hex file, probably INTEL HEX format), then run through a disassembler program to generate a source listing in Z80 assembly language.

At this point you may get lucky - a "hex dump" of the file, separate from the disassembled listing, will probably show you some ASCII strings - text that give you some clues what it does.

(Same is true for whatever other microprocessor was used ... if the EPROM stored the configuration for an FPGA you have a whole different problem).

Then comes the difficult bit ... understanding what it does, well enough to successfully modify it. 

Even if it was originally written in C or another language, you're not, realistically, going to be able to recreate anything higher level than assembler from it.

Knowing what it does will also require intimate knowledge of the hardware design : for example, if it writes a number to an I/O port address, you need to know what hardware is connected to that port, and what that number means to that port... for example, if a UART is mapped to that address, it'll probably be mapped to 4 addresses, and one of those addresses will be its "Transmit Data" port. Then whatever number (say, 65) is written to that port will appear on an RS232 serial port - in this case as the character "A". (If it's written to the UART's Control port, read the UART's datasheet : it could disable the UART, or change the baud rate, or do something else altogether).

If you have all that knowledge then you have a laborious and tedious but possibly feasible task.

If you don't, (and possibly even if you do) then it is probably faster and cheaper to define exactly what the circuit does, and re-implement it from scratch using whatever microcontroller you are most familiar with (Arduino etc) - in C, C++, Ada etc.