How to unlock Eve’s Dead Bird

Turns out the Gel Canister is also available on the Score Attack mode. It's unlocked after the Arsenal is built on the Bastion. After getting twenty kills with one turret from the Gel Canister secret skill, the achievement was awarded.

First you have to own Flash; he's the robot Dog that costs 8,500 coins in the Gadgets menu. The achievement is only possible with Flash equipped to a Gadget slot.

All you have to do is travel 10,000 meters cumulative with Flash equipped. You don't have to reach 10,000 meters in one run, multiple runs still count toward the total. After you've gained 10km in travel with Flash you'll be awarded the achievement upon your death.

There's a counter in My Profile under Gadgets that lists "Total Distance run by dog"; this is the count that matters. The counter appears to go up whenever Flash runs, not Barry. I'm not sure if it goes up when Flash jumps yet.