FWIW, our liftgate started working intermittently when the battery started to go bad in December 2010. So if you haven't replaced your battery yet, I'd at least get it load tested at your local auto parts store. If it is bad I'd strongly recommend only replacing it with an AC Delco p/n 47-7YR (same as factory installed battery). I say this because the factory battery is rated for 650CCA while all replacements I've found are 600CCA or less. I see you're from Canada so you'll want the most powerful battery you can find. Oddly enough, this is one of the very few applications in which the factory-equipped battery provides the most power.
If the battery checks out OK....
I have attached a diagram for the liftgate switch circuit for you below.
2007 Equinox liftgate switch circuit.pdf
It appears that the latch is activated when the liftgate switch grounds out the "Rear Closure Handle Open Signal" wire going to the BCM. That tells the BCM to output power to the coil for the "Liftgate PCB Relay" which closes the relay and allows power to flow from this relay to "L/Gate Fuse 4".
You'll need a helper when testing the circuit for fuse #4 because the power wire terminal at fuse #4 will only be hot when the switch at the liftgate is being pressed (assuming the liftgate switch and BCM are functioning correctly).
If you get power at the fuse terminal while someone is holding the switch, then you likely have a bad latch.
If you don't get any power then you need to try to bypass the liftgate switch by grounding out the "Rear Closure Handle Open Signal" wire. If the latch works when the switch is bypassed then obviously the liftgate switch is bad.
If when bypassing the switch you still don't have any power at the fuse, then the "Liftgate PCB Relay" is likely bad. Unfortunately, it is not serviceable and would require replacement of the entire instrument panel fuse block since this relay is integral to the fuse block.
If one were skilled enough, it may be possible to bypass the internal relay and correctly install an external relay to function in its place. This would obviously be a much less expensive option if necessary.
To get to the back side of the fuse panel you need to remove the console and the panels on either side of the shifter tower.
I've also attached instructions for replacing the IP fuse block below.
IP fuse block replacement.pdf