I know this is an old thread, but people still look for answers here and I have just had a go at an SQAi body that was looking in a similar state to the OP and I actually found a solution (at least to mine) so I thought I would share this in case it helps someone - not power or lights on the SQAi body and thus no shutter/mirror firing.
The solution - The two contacts beneath/inside the body for the battery compartment to connect to had some oxidisation on them and didn't make proper contact, which was the first problem. There are 5 screws to drop the bottom plate off - 2 at the rear and one at the front of the bottom plate and then 2 more underneath the textured rubber by the tripod screw. Once removed the plate will come off, but watch for the spring loaded black pin by the one front screw (by the motor drive connection pins) as this can spring out. Once inside you can see the two small connection pins at the back of the battery compartment and the two flat spades on the circuit board that they are supposed to contact - cleaning both these with petrol/alcohol and also just ruffing the metal surface regained the battery contact and the lights came on the camera again. I suspect this is the most common battery fault here and adding a piece of card in with the batteries as people have described above would put a bit more pressure on these internal contacts, but if it is really oxidised, will not make the difference (I tried the cardboard first and nothing)
So lights are on now, but shutter still not firing. I actually looked at the manual for this one and in section 13 it said the light will come on if the shutter button is pressed, but it still won't fire due to several states the body may be in - in my case it was 'the lens release button is depressed'!! For some reason on my body which I haven't figured out, the lens release button does not return to its home position and stays semi pushed down - if I just manually push it back (up) to the home position, the body fires and works as normal. I don't know if this is a common problem also, I have tried a small drop of oil etc, but it is still stiff to return - job for another day I think.
Any, a combination of working out both issues has returned a dead body to life and if your SQAi body is in a similar state, this might help you.