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Having completed this Easter Egg, I can tell you that the best strategy (which is also stated in this guide) is to:
Run in a circle around the Cosmic Way Portal (the one that takes you to Pack-A-Punch)
Run over the upper bride that leads towards the spawn area
Run under the lower bridge back towards the Pack-A-Punch portal
Repeat
By doing this, you will eventually gather all the zombies into a nice trail behind you, so you won't have to pay attention to them too much. Whenever possible, shoot the Alien while performing the above steps. I don't remember if there was a way to tell how much health he has left, but the linked guide above states:
The indicator on the boss goes from green, to yellow, to red so you’ll know how much health it has left.
It's important to note that you must finish the Alien with a melee attack to his back. After so much damage, he will kneel down. This is when you can perform a melee on his back to get the Alien fuse.
Other tips/things to keep in mind:
A Wonder Weapon will help you greatly during the boss fight. One is also required to destroy the Alien Ship and thus fully complete the Easter Egg (see the guide linked above on how to do that). Personally, the Face Melter is the best.
The pistol you start with (Kendall 44) is actually an extremely good weapon after being Pack-A-Punched. The upgraded version called Baby's Breath, is similiar to the other Zombie games upgraded spawn pistols. You will get an akimbo variant that shoots grenades. This, in combinations with the "Bombstoppers" perk, makes it a deadly combination for taking out a large horde of zombies. You can even shoot danger close to yourself and not be harmed so long as you have Bombstoppers. If you can, put extended mags on this pistol as well to get more ammo with it.
One Alien Boss spawns per player in the game. However, if you have a friend or friends help you up to the point before you spawn the Alien, if your friends quit out, you will only have to fight one Boss. Make sure that you are the one hosting the game, since if the host quits, the game will end.
Don't forget Mule Munchies so that you can have three weapons! You'll want all your weapons upgraded.
Having either a Boombox or a Sentry Turret from the souvenir machines on standby is always nice. I would go for a Boombox as this acts like a Monkey Bomb and distracts zombies for a significant amount of time before exploding and killing them. It could be a real life saver. Since one of the souvenir machines is right next to the Pack-A-Punch portal, it's a good idea to have one Boombox on you and two of the three coins already deposited into the Souvenir Machine. Keep the third coin on you, and when you need another Boombox while fighting the boos, throw it in the machine and get another.