I got my dog from someone who said I could have one if I gave them riding lessons up to the value of the dog. We signed a contract stating this. Unfortunately the horse she would be riding died. I worked for this person getting paid well below minimum wage and after looking at hmrc found out this was illegal so said to them that we would call it quits and not pursue money owing to me.

Sure they can. YOU agreed for something in exchange for something else... just because the horse died doesn't end the debt... give the dog back or find some other compensation they agree with.

IN the USA farm workers are one of the jobs not covered under the minimum wage act... if you work with Horses... you are a farm worker.

You have a contract stating you would be paid in exchange for providing a service. You haven't provided that service, so therefore you shouldn't be paid. The fact that you were paid in advance means that you must return that payment (the dog). However, if you are willing to bargain away back pay you believe is owed you for the value of the dog, and if the other party is amenable, great, no problem - just make sure to get it in writing. But if the other party is not amenable - i.e. they believe they do not owe you for being paid too little - then be prepared for a possible lawsuit to force you to return the dog, and then you countersue for back wages..

Are you sure that your job is one that is covered by minimum wage laws? What type of job was it? Were you an employee of this person, i.e being paid an hourly wage? Or were you a contractor providing a service?

Courts don't allow one charge to be part of another, in general. You signed a contract that had no connection to the work you did. You have to give back the dog and pursue the other case separately.
If you were paid below minimum wage I have a feeling you have not reported it on your taxes? I am not sure you have good grounds for a trade-off. Many casual jobs are not reported by either party, a good deal for both, because you don't have to deal with money withheld. Babysitting, petsitting, occasional house or yard work, etc.

Some intelligent negotiation is called for here so you can keep your dog, whom I am sure you are in love with by now!

Sure they can. YOU agreed for something in exchange for something else. ...

No, not unless they perfected a security interest in the dog, which is highly unlikely. Their remedy would be the money value of the dog, not the dog itself.

In NY this would boil down to the dollar vlaue of the dog. The dog would not be "repossessed."