What are the requirements on buried propane lines?
I cannot fully answer your question, but it seems that New Hampshire simply uses the ICC guidelines for building codes, including for "Plumbing, Mechanical and Fuel Gas" which seems to be the key area here. 

Not being an ICC member, I can't access the documents, and neither the New Hampshire Statutes website nor the State Building Codes website reprints these documents for public viewing. This may be to encourage the use of licensed individuals for work subject to code, but IMO as "ignorance of the law is no excuse" in most jurisdictions you should be able to educate yourself on the building code free of charge, the same as for any other section of law. I would call your county clerk and ask how to get ahold of a copy of the codes. It may cost you a little money for their time and paper, but if they can help you here it probably won't be as expensive as an ICC membership for yourself.

If the government won't help (excuse my cynicism but I'd put money on various offices passing the buck), if you need chapter and verse, just get it from both the propane guy and the gas company. They're telling you two very different things, and they can't both be right. Ask both companies for a hard copy of the section of the building codebook, with title and year of that document, detailing the relevant codes which back up their understanding of how it should be done. Both are working for you, and one has shafted you; my money for which one is with the one who can't quote the most recent official codebook recognized by the State of New Hampshire.