I want to install a water softener for our house and the company that I am dealing with told me that the unit must be installed outside unless I only want to soften the hot water. Is this true?
The main water line will go in the house far enough to get a good spot to tie in a water conditioner. I had mine in the crawlspace, it was 3' tall and I dug it into the dirt to get the height needed to fill and maintain it, also to protect it from freezing (Maryland at the time). Since I did not want it in the living space taking up room, that is why the crawlspace. Most conditioners are placed in the living space perhaps the laundry room, or at least the garage, depending where the water main is accessible. Any competent installer can run the pipe where it is needed to install the system.
My suggestion is to get or try another installer. Always good to get more than one estimate. My conditioner was for the whole house, not for just the cold or hot. I did not have the outside hose bibs tied to it, no need for that.
Based solely on the description I would look at it that the water softener to feed all of the cold water for the home has only one common point that is reachable and that is where the pipe comes into the home. As it goes into the house it branches and any place assessable before that branching is not found in a location that is able to support the install without opening walls and running a loop off the main line.
Where as the hot water heater is inside and has a place for locating the softener but this locations cold line only goes to the water heater.