It's not a good idea to lay them on their back. Ever try to swallow something laying down?
There's a good article in the stickies for Orally medicating your Bird.
I do it the same way my avian vet does, which is holding them like a baby. Although the back is down, they are not in a completely supine position, since the head is higher than the rest of the body.
You're dribbling it in a little at a time, not squirting it in - not any different than drinking from a straw while propped partially up in bed. I can tell you that, at least with my guys, if you held them upright, they would be spitting it out faster than you could get it in, unless you had a third hand to tilt the head back. Also, no way that I could medicate a large macaw by myself while holding him or her in an upright position, without risking injury to the bird - they are pretty powerful birds.
ETA: When it says "with food", it doesn't mean it needs to be, or should be, mixed into the food, just that it should be given around the time the bird is eating, so that s/he doesn't get it on a totally empty crop. IME (with three different macaws, a duck, and chickens), they detect even the taste of Metacam, and won't eat the whole of whatever you've mixed it into.
