Where is it coming out of?
You know, my instinct is? Vet.
Very many things can be handled at home. But if this is seriously a vein or artery? You don't want to go pulling, poking, pressing, or trying to repair it.
And this begs the question, how?
Lots of things can LOOK vein-y: sinew, a tendon string, a muscle fiber; a vein? Would, generally, be bleeding. Badly.
IN any event, ANY of these things; any scenario of the sort you describe? Sound scary to me; abnormal, and if there's a puncture or rip?
This may be the first of bad things to come.
I'd urge an abundance of caution; if you can't find a vet open, at least an experienced Chicken-keeper.
My experience is not with chickens, but I'd make haste.
I am NOT a medical practitioner or vet.
I'm new to the forum; I do know you can post photos if you publish it to your desktop or know the file number; but I'd be glad to give you my email in a PM, if you can send it that way.
I can not promise anything, but I'd take a look.
Again, if this IS a vein? I'd suggest a vet, and quick.
Emergency vets are uber expensive, but often they WILL accept a photo by email, too; to help give you clarity.
Please ;let me know what you were able to do; O, and? I'd immobilize and isolate, if you have not done so already.
Please let me know...
Thoughts are with you.