No, you do not.
There is absolutely no rule, law, requirement, or regulation under the Jones Act or general maritime law that says you must give a recorded statement after your offshore accident. Because you were hurt on a tug boat, you can take advantage of these provisions of the Jones Act, which applies to any injuries on vessels like tug boats, barges, dredges and supply boats, and also to floating rigs like jack-up rigs.
This may come as a surprise to you. Maritime companies often tell their employees after an offshore injury that they must give a recorded statement. The companies tell injured offshore workers that without a recorded statement, their employee is not eligible for benefits following their injury.
This is not true. If you are injured on the job offshore, under the Jones Act you are entitled to "maintenance & cure" benefits. There is nothing in the law that requires you to give a recorded statement in order to receive these benefits, or to receive any of the other benefits available under the Jones Act or general maritime law.
You may be wondering why your company wants a recorded statement from you? The answer is simple - they are trying to shield themselves from liability for your injury. You see, the company will send a professional claims adjuster to obtain a recorded statement from you. These adjusters are trained to take a statement from you in a way that will help the company, and hurt your claim. The statement is recorded, so if you ever get cross ways with your company about your benefits, they will throw your statement in your face, and deny your claim.
If you company refuses to provide you with benefits unless you give a recorded statement, they are breaking the law, and you would be wise to consult with a lawyer who has experience in handling Jones Act lawsuits.