How to program a BeagleBone board?
The processor on the BeagleBone is not a microcontroller, but rather a microprocessor (the Sitara AM3359 from Texas Instruments), which doesn't have non-volatile memory on the processor itself, but rather tries to load programs (such as an OS) into RAM via a communications interface (UART, USB, Ethernet) or from an external memory (FLash or memory card). The initialization/booting process for this processor is described in detail starting at page 4667 of the technical reference manual.

It seems the latter (booting from a memory device) is the case for the BeagleBone, which means you can "program" the BeagleBoard by changing the content of the on-board Flash (eMMC) or by forcing it to boot from the microSD card. It's not very clear from their documentation how easy or hard it is to create your own memory images (for SD card or the eMMC). I suspect it's not trivial.