I am wanting to do the following, however, when I do it, it replaces the fence the chicken was standing on with air. I figured a simple solution to this would be to replace the entity instead of the block.

Before:

otherdrops: 
    CHICKEN:
      - tool: MONSTER_EGG@0
        action: RIGHT_CLICK
        consumetool: 1
        replacementblock: air
        flag: UNIQUE
        drop: MONSTER_EGG@THIS
        message: ["&bAll right! &bYou caught &c%v&b!"]
      - tool: [ANY_TOOL, ANY_OBJECT]
        drop: default

After:

otherdrops: 
    CHICKEN:
      - tool: MONSTER_EGG@0
        action: RIGHT_CLICK
        consumetool: 1
        replacementmob: air
        flag: UNIQUE
        drop: MONSTER_EGG@THIS
        message: ["&bAll right! &bYou caught &c%v&b!"]
      - tool: [ANY_TOOL, ANY_OBJECT]
        drop: default

I thought I had made a ticket about this, but I guess I only mentioned it on the main page. You can use "damage.victim: 100" to remove the mob without removing surrounding blocks. The problem is that they will then drop their default drop along with your custom one.

We discussed adding something like damage.victim.remove to solve the problem, which I think is a better alternative.

Fixed in d1d18a7. replacemenblock: AIR on an entity will now only remove the entity - other replacement blocks work as usual, as does replacementblock: AIR on a blocktarget.