Your question is a good one because when people stop paying for their mortgage, for whatever reason, the bank--who loaned them the money in the first place is now no longer their friend, and now these borrowers who were once happy to get a loan so they could buy a house, feel their house is being stolen from them. It's human nature.
But unfortunately the answer to your question is NO, you will be expected to leave all the various parts of the house for the next owner.
The fence is a part of the home as much as is the landscaping including plantings, walkways, the garage, and other interior items like carpeting, appliances and so on. In Florida we are seeing homes going to foreclosure with items missing--like air conditioners, pipes and wire. As it turns out, that falls under the bank fraud statutes and will involve the FBI. So a simple fence that you installed and then took along with you can turn your foreclosure into a jail term.
Sorry to share reality, but that's how it is.
Your contract terms in the note state you must maintain your property and violating those terms could cause them to accelerate the loan but you already violated those terms when you stopped paying so, while you are in breach of contract for taking the fence, you haven't committed any civil or criminal offense unless you have some local or state ordinance/statute/law to the contrary.
I'm not an attorney (Thank God) so, this is my opinion, not legal advice.
Buy a home after foreclosure expert
Just from past experiences, and I've been involved in many where the homeowner took things including fence, sidewalk, landscaping bricks, etc. It is best to leave these things if they were there when you purchased the home, or paid for out of funds used from a HELOC. The reason for this is it can be considered intentional damage, and in certain states it is treated as vandalism. I know of 4 homeowners who did jail time for things like this, so be careful in what you are thinking of doing. Do the right thing.
My question to you, is, why not do a short sale instead of just letting the bank take the house back. That will let you stay in your home a few months longer & keep the foreclosure off your credit. Contact me if you have any questions or you want help doing that.
Terry Bell, Realtor, Santa Rosa, CA