Okay so say you do multi-class with the AP. At 5th level AP, you get fiendish boon and the possibility of choosing a companion instead of the weapon. Now if you multi-class, it will be too weak to ever really help much out in combat other than a diversion or speed bump. However, would the Boon Companion feat help with this?
Boon Companion
Your bond with your animal companion or familiar is unusually close.
Prerequisite: Animal companion class feature or familiar class feature.
Benefit: The abilities of your animal companion or familiar are calculated as though your class were 4 levels higher, to a maximum effective druid level equal to your character level. If you have more than one animal companion or familiar, choose one to receive this benefit. If you dismiss or lose an animal companion or familiar that has received this benefit, you can apply this feat to the replacement creature.
In all intents and purposes, the AP fiendish boon is a companion and a class feature, much like the wizards familiar and rangers animal companion.
If you feel this would not apply, are there any alternatives?
Anti-Paladin doesn't work like normal Animal Companions. Normal Animal Companions all go off of this list. while the Anti-Paladin gets a permanent monster off of Summon Monster spell list.
The Anti-Paladin's Fiendish Boon is in no way an animal companion and would not be helped by gaining Boon Companion.
Since the anti-paladin companion is treated as a summoned monster not an animal companion or a familiar then no, boon companion doesn't help.
However, anything that affects summon monster would. So you could take the feat Evolved Summoned Monster and apply it to your companion. Basically anything that summoners might take that has to do with their summon monster ability, anti-paladins can get and apply to their companion.
If it's for a home game it's worth asking your GM to permit it. While RAW it doesn't apply, as a GM I'd allow it to work, on the grounds that it's in the same spirit as companion/familiars and not OP.