Sounds like the old stuff may be Transite® or cementitious asbestos board (CAB). this was often used in that era as both shingles, and as skirting. It should be easy enough to safely remove, but I would strongly recommend you get information on how to do this correctly and/or legally. I don't know what the laws in NH are but in WA you can remove it yourself with a permit.
As for a replacement, the closest thing these days is a tile backer board, but I would not recommend that. The CAB was great in this application except when it got hit with something. Try Hardieboard
On a job we once removed well over 100 4'x8' sheets of CAB. We took them off in whole pieces. What a waste, would have been good for another 50+years