Your question depends a bit on what was done when the aftermarket stereo was installed. Standard aftermarket car audio systems are quite simple... the head unit (your cd-player/radio) does all the processing and amplification internally, then sends four (Front L&R, Rear L&R) speaker level outputs that you wire directly to passive drivers in your doors and rear panels.
The Bose systems (at least the one in my 92) aren't set up the same as most standard car audio systems. The head unit feeds an outboard processor which then sends line level signal and amplifier power to each of the 4 individually amplified speaker enclosures. The Bose amps are built in to each enclosure along with the driver, and it's at the speaker location that the signal is amplified.
To use the bose speakers with an aftermarket head unit, he would have either had to bypass the bose amps and send speaker level directly to the drivers, or somehow send line level to each amp. Do you know if the system was ever functional?
Assuming you're not concerned with keeping things original, my suggestion would be to replace the whole mess (those Bose systems didn't exactly sound great to start with) with decent aftermarket components and run speaker wire to the drivers. You should definitely test the head unit that you have before you replace it though -- it may be fine, just wired incorrectly.
Hope this helps...