The only way I can think of to reduce the reflection from the eye, is to use flash set away from the camera so the light doesn't bounce straight back to you. I doubt that you will get rid of it entirely though. You could always fill with black later in PS, I suppose. Personally, I wouldn't worry too much about it, it is natural for critters at night to have shining eyes as light hits them, so it doesn't look too odd, as we naturally see them like that with our eyes. It isn't as bad as 'red eye' in portraits, where you capture something that the viewer will see as unnatural, because red eye doesn't get seen normally.
I would suggest that you use caution about flashing the squirrels in flight. Whilst the flash won't physically harm them, they will be temporarily blinded by it (just as you would be if adjusted to night vision and somebody set one off in your line of sight). If they are gliding and suddenly become unable to see for a minute, they are likely to crash into the tree they were planning to land on, or possibly even miss it completely (or hit a branch) and take a tumble to the ground. They are pretty resilient critters and should still be relatively unharmed by a fall, but there is always a risk they could break a leg and that could hamper their chances of survival.
Flashing them whilst sat in a tree though won't do them any harm, they should have enough sense not to take flight when they can't see where they are going.
As for settings with Av and Tv, you are going to have problems in that light to get anything. If you are handholding you will be struggling to get a fast enough shutter speed to avoid camera shake, if on a tripod you still need a reasonable shutter speed to avoid motion blur if the critter moves. In order to get the fastest shutter speed you can, without risking underexposure, I would suggest you use Av set to the widest aperture you can and set a high ISO. Then see if you have enough shutter speed for the lens you are using (if you are at 300mm, ideally you would want 1/500th on a cropper, but can get that down a bit with good bracing technique. IS will also allow slower speeds handheld of course, but if the squirrel is moving around you will still need at least 1/250th and preferably more (even at that, you would need to time your shots to when he was fairly still, although a pan shot as he glides would work OK with practice).
P.S. I'm jealous you have such lovely critters in your back garden.
