Generally, calls in the back country are rare due to the fact that there is no population there and most of the people who to hiking/backpacking back there know what they are doing and are in generally good shape. They do happen, but the call volume is so low that it is exceedingly rare that there is EMS assigned to that area. Now the less back country but still pretty remote campgrounds of many national forests and national monuments, especially in the Western United States, have their share of medical and trauma calls where the patient is significantly into the forest, down a hiking trail, away from paved roads, down steep hills, across a river, etc. The people getting hurt there tend to be less experienced (and often times drunk) so there are come good calls out of those campgrounds.
If you want to try out that type of EMS, and while it is not exactly back country it is still fun, consider some of the local EMS agencies that cover those areas. National Forests do not have their own EMS. (larger National Parks do.) While there is forest service law enforcement and USFS Fire to first respond, they do rely on the local EMS to respond to calls and provide ALS. One of the companies I work for covers over 1,000 square miles of the Sequoia National Forest and the other covers several hundred square miles at least although I have not seen the exact stats for that one. I have been on numerous hike in rescues where we have the patient for multiple hours either hiking them back out, or waiting for a hoist rescue. On one occasion the patient was in an area where we could not hike them out and because of inclement weather we could not get a hoist rescue until the morning so we spent the night with the patient. If being a ranger is your dream, by all means follow your dream. However, the local EMS agencies who cover the forest might be something to consider in the meantime.
As far as my area in California goes.
Kern County: Liberty covers parts of the Sequoia National Forest, Sequoia National Monument, and BLM's Chimney Peak Wilderness. They also cover several small mountain communities so their entire response area is rural.
Tulare County: AMR Porterville and Imperial Ambulance both cover parts of the Sequoia National Forest and get a decent number of calls up there in the summer especially. (They also cover a medium size city so most of your calls will be there.)
Fresno County: Sequoia Safety Council covers parts of the Sierra National Park and King's Canyon National parks and the surrounding wilderness areas.
Madera County: Sierra Ambulance covers the Sierra National Forest and parts of Yosemite National Park. They are well known for their long transport times and hike in rescue calls. They are an extremely rural provider.