How do I summon a chicken zombie jockey with diamond armor using a command block?

You want to use the Riding: attribute when you spawn it in. This will produce a zombie chicken jockey in full diamond armor, with a diamond sword (the first item):

/summon Zombie ~ ~1 ~ {IsBaby:1,Riding:{id:Chicken},Equipment:[{id:276},{id:310},{id:311},{id:312},{id:313}]}

It's very likely that you just need to add Riding:{id:Chicken} to what you've currently got in your command block, and adjust the coordinates to fit.

The decimal values for armor and swords (or other weapons, etc) can be substituted to fit whatever need.

Editorial note - reader beware! It is extremely fun to do positively silly things with this and then realize you spent the whole day doing it.

I would use this:

/summon Zombie ~ ~1 ~ {CustomName:"Sittin' On The Chicken",
CustomNameVisible:1,IsBaby:1,Equipment:
[{id:diamond_sword,Count:1,tag:
{ench:[{id:16,lvl:1}]}},
{id:diamond_boots,Count:1,tag:
{ench:[{id:0,lvl:1}]}},
{id:diamond_leggings,Count:1,tag:{ench:[{id:0,lvl:1}]}},
{id:diamond_chestplate,Count:1,tag:{ench:[{id:0,lvl:1}]}},
{id:diamond_helmet,Count:1,tag:{ench:[{id:0,lvl:1}]}}],
Riding:{id:"Chicken",IsChickenJockey:1}}

Try this command:

/summon PigZombie ~ ~ ~ {Equipment:[{id:276},{id:310},{id:311},{id:312},{id:313}],CustomName:lol Zombie,Riding:{id:"Chicken"IsChickenJockey:1}}

To summon a baby zombie with items, type this:

/summon Zombie ~ ~1 ~ {IsBaby:1,CustomName:Equipped Baby Zombie,Equipment:{[id:297]}]

and bam, there you go. Copy and paste this on your Minecraft server.

OP Zombie Villager Chicken Jockey Command:

/summon zombie_villager ~ ~1 ~ {CustomName:"Your Worst Nightmare", CustomNameVisible:1,IsBaby:1,Equipment: [{id:diamond_sword,Count:1,tag: {ench:[{id:sharpness,lvl:10}]}}, {id:diamond_boots,Count:1,tag: {ench:[{id:protection,lvl:10}]}}, {id:diamond_leggings,Count:1,tag:{ench:[{id:protection,lvl:10}]}}, {id:diamond_chestplate,Count:1,tag:{ench:[{id:protection,lvl:10}]}}, {id:diamond_helmet,Count:1,tag:{ench:[{id:protection,lvl:10}]}}], Riding:{id:"Chicken",IsChickenJockey:1}}

Use this for good enchanted gear drops.