Should I defend the Entry Room?
Option 1: Defend the Entry Room.
Doing this does two things: reduces the damage to the rest of the dwellers in the other rooms; and increases the time it takes for invaders to reach the other rooms.
This can reduce the overall time that your Vault is "under siege" and losing productivity, as well as give you extra time to prepare (you can move Dwellers around, under siege, and they will automatically return to their rooms once the siege ends). However, Using a Mr. Handy purely as a defensive strategy is a bit of a waste, as they also have the "auto-collect" feature, which is very handy, and is more useful on a floor that actually has production rooms.
Option 2: Don't defend it.
Doing this means that the intruders immediately move into the Vault. Depending on your layout, this can draw out the overall time you are under attack, as well as creating the possibility of killing off your valuable workers, reducing morale, and losing productivity.
This is where your Mr. Handy is more useful. As well as defending the entire floor (he will follow the invaders through the entire floor), if he is destroyed, he will not cost any morale issues if you are incapable of reviving him.
EDIT: Dwellers do keep working until the attackers reach their rooms. You can still collect resources while the room is under attack, but they do keep working if their room is not under attack. (Thanks to OrangeDog for confirming this)
My answer would be "No."
A downside to defending the door room not mentioned so far is that the guys guarding the door aren't doing anything else. Whatever productivity you may gain from having raid/deathclaw events end a few seconds sooner will be massively offset (in my view) by having two high-spec dwellers spending most of their time kicking their heels in the entrance room instead of patrolling the wasteland or questing or crafting or producing stuff (or getting ).
If your vault is so well staffed that you can spare two people to do that, then go ahead, but otherwise it's a waste.