Before the point by point stuff, Saxton's insistance that shields need to be lowered is because of the Heir to the Empire boo or so, by Zahn, where the shields of Coruscant prevented comms.
The occasion was to good not to catch to pretend that ISDs had their shields lowered.
If i got it correctly, the shield blocked Holo FTL transmissions only. I'm not even sure it was specific to holocoms in fact.
This needs to be clarified beyond any shadow of doubt.
I've wondered how then could the Executor even KNOW that the Emperor was trying to contact Vader. They obviously were capable of receiving a form of FTL message. What would be so magic about holos that the signal couldn't be boosted, with some packets eventually made redundant to account for packet loss?
I've always thought, rather logically and obviously, that Vader moved his ship out of the asteroid belt to obtain a clean communication, 
regardless of the shields, but simple because of interferences due to the asteroids, notably by a signal coming a long way since some world in the core I suppose.
That's also how it completely hurts their whole argument, and why they insist there was a gap in the Hoth shield, to let machines walk under, while we know that if it's anywhere near to the Gungan shielding, such a gap would not be necessary.
Are we going to argue that Gungan shielding is superior? Huh.
Besides, why would the rebel have a gap in their shield? If they need to communicate over long ranges, they can have secret comm systems outside of the shield perimeter.
Jedi Master Spock wrote:
3. Order 66 went out by FTL holo-communication and did so through active shields, with no visible problems, in ROTS.
I remember asking there or at SBC how the ARC-170 got their comms through their shields. We assume shields are up, since they're in a battle context and flying close to enemy positions.
But there could be a relay negating the FTL side of it.
I don't remember any other evidence of a holo-communication event with shields clearly surrounding the receiver's communication system.
I mean, you may eventually find a reference in the novelization, but would you have the proof that a whole shield protected the vessel, vehicle, base or station, and that no antenna was sticking out of the shield?
4. The difference between voice communications and holographic communications is just bandwidth. At the resolution seen in SW movies, it's not even a terribly large amount of bandwidth. Communications go between shielded ships all the time (see attack sequences) and even through the very powerful Rebel shield on Hoth.
Clearly, there's no reason in theory why holo data, or FTL holo data, would be entirely blocked while all other types of comms make it through.
In ANH, the rebel fighters could communicate lively with the Massassi temple, yet they had to get through two types of shields.
5. Shields are selectively transparent to certain frequencies/energies, hence why you can see through them most of the time. Again, no reason why this can't be used for a communications channel.
On this I'd rather argue, for the sake of all TV SF as a whole, that there's a threshold, and that only a certain amount of light is allowed through. Obviously this argument is limited, but it's better than nothing. That said, I'm certain the amount of light allowed through shields would easily be enough for holo comms.
But light is not a FTL medium for comms.
6. Shields don't appear to have anything to do with FTL comms. The basic principles suggest they wouldn't interact in any event.
This is why we really need to get that reference from Zahn's book right, to at least understand part of the origins of the claim.
We'd also need to get the line from the ICS after all.
And finally, problem number seven: 
7. There's something of a lack of evidence for lowering the shields. Thus, even without 1-6, it's a flimsy explanation at best.
It's also how SW ships have patches of shields, and thus certainly not require to get the whole shield protection dropped.