Here's a long thread that discusses limiting a renter's ability to write a negative review.
Does your contract specifically state there will not be a refund in case of a natural disaster? Does it define natural disaster? Did your guest sign the agreement?
There has been some discussion about owner's refunding rent in case of natural disaster despite a written contract indicating otherwise. Many owners thought it was the right thing to do, depending on the severity of the disaster and how it impacted the services at the property. (i.e., water, electricity, accessability, etc).
And, there was discussion about researching local regulations on whether a property may be considered habitable under these circumstances.
Search on "natural disaster refund" for more information.
I hope you are able to work out this situation.
It's always hard to respond to this type of post because we only get one side of the story (either owner's or guest's)...and not enough details. There could be a lawyer who might want to chime in, for exampe, but can not for the lack of detail. I am not a lawyer. Have you contacted an attorney?
I live in an earthquake / tsumani region and understand disaster risks. But, it's difficult (for me) to grasp fully what actually went wrong at your place with your natural disaster. I'm wondering what was the extent of the "damage" directly put upon the guests?
Was there a hurricane? Was there official evacuation by your State? Did they have to go to a shelter? What happened to the surroundings outside your home...the entire location....were "parks" closed that your guest traveled from afar to experience by staying in your home? Was the area flooded?
Would you give us more details, please, about the extent of the damages specific to your home caused by the disaster?
How does the guest think it impacted the quality of his stay? For example, were all the "beaches" closed ....?
As 'thaxterlane' remarked, it's about the severity to the surroundings, house amenities (like a swimming pool that became unusable, etc.), bad or rationed water supply, limited or no house utilities..etc.
Did you have a rental agreement with a cancellation / refund clause ref. weather, natural disasters, signed by the guest?
I, otherwise, can't respond with how I would handle this "threat" of lawsuit + a bad review. I would like to know why you think a full refund under those particular natural disaster conditions shouldn't be given....and why your guest demands it?
Personally, because I live with "disaster threats"...: sometimes, it is the right thing to do for innocent guests caught in a bad situation vacationing.....even though we might have a contract "weather / disaster": clause. Sometimes, the compassion we extend to our guests works for us to far outreach what we gain by holding onto their rent money. Especially in the face of a lawsuit (which might be just a threat...but maybe not)...it's not worth it for an owner who could end up paying more than the rent you gained + damages awarded to the guest + your having to pay all lawyer's fees - both yours and the guests. Think about this: your guest could be uploading the "mother of all wonderful reviews" about how well the owners managed everything under a natural disaster, instead of "flaming" you + suing you.
I think that the best thing would be for you to contact a local attorney to draft a release of liability form for you to use which includes a clause that the guest will not post a review of your property. This will probably cost you a few hundred dollars in attorney's fees. An alternative would be to search on-line for release of liability agreements. Laws vary from state to state. Look for forms for the state in which the renter resides and also the state in which your property is located and compare them. (Hopefully, they will be similar.)
Did the guest ever actually stay at the property? I believe that under HA/VRBO policy, if an individual did not actually stay at the property, they cannot post a review.
To answer the OP's question, I don't think the owner can require the renter to sign an agreement not to enter a review to refund their money if they were in any way obligated to make the refund. If the refund was in any way warranted, then adding extra conditions to its payment would put you on shakey ground.
The question now is whether you were obligated to make a refund, or whether it was out of the "Goodness of your heart" that you cut the renter a break.
I'm not a lawyer, but have studied the topic for our own contract. Here is the refund criteria for natural disasters and general unavailability of services that I uncovered.
Most of it has to do with whether the product or service offered was available as "Fit for the purpose intended."
So, if the house was unreachable due to high waters, it obviously was no longer fit for the purpose intended. Likewise, if there was significant water in any habitable area of the house (basement bathroom), most would consider it unfit for the intended purpose. If power was out for an extended period of time, even though an "act of God," it would likley be considered unfit for the purpose intended - i.e. sleeping, waking, cooking, watching TV, reading after dark, etc.
If it were only that some of the features were unavailable (no cable TV) but the hose could still be used for daily living, it would still be fit for the purpose intended, albeit you may feel ethically obliged to offer a partial refund. Same for any other feature that is not necessary to live in the house but makes it unique or adds special value.
And at the other extreme, if the flood closed down all the local parks and shopping malls, but the house was untouched and reachable by road, I wouldn't believe you had any obligation to offer a refund - that's what trip insurance is for. Any refund you give at this point would be solely for brownie points - they should give you a glowing review if you were to give them something to offset their bad luck.
For us, we have a clause that states we only give weather refunds if the state police close the interstate exit by our house. We also included a specific clause that says if any feature (hot tub, PS2 console, gas grill) is out of service, we will refund $20 per day for each feature unavailable - this sets the ground rules right away so people don't start asking for the sun and the moon if something isn't working during their visit. It also gives us a chance to jump in right away if we find something is unavailable to them and tell a guest that we will be giving them a refund - even if they didn't ask for it: this is serious great-review stuff and has led to at least 2 of our 5-star reviews.
I know only part of the above may be applicable in your situation, but regardless which part might apply, I hope it is helpful!