Bodhi (and any linux OS) can be installed to any drive, including external drives.
What likely happens in your case, is that the bios has not detected the usb yet when you select the bodhi menuentry to boot.
Try this out.
When grub menu appears, go to prompt (press 'c'), then type in "ls", see if the external pendrive is listed. If it is not listed, at that point, it will not boot any OS in pendrive.
Type again "ls", when pendrive gets listed, then OS's in pendrive will boot.
It's just that some bios take some time to detect drives, usually external usb drives.
Another way is to wait longer for bios before booting up.
Firmware updates for bios can help. Sometimes changing to other usb ports can help (or just cleaning the contacts). Or (hopefully not in your case) a better usb drive (try fsck'ing the usb partitions first).