I have a lot to say about this issue because I was in this same position a couple of years ago. I was working remotely for an overseas marketing agency, and at the beginning, it was fine. I had no major complaints, although some procedures were very off-putting, but I had worked for worse places, so I didn't think much of it. After six months in, the owner of the company sent an email informing us that we were going to download tracking software that was going to measure our productivity and other stuff.
I've heard of these programs, and I knew that many employers loved them, especially when having remote employees. The problem was that the software that they wanted us to install went beyond tracking our work… Firstly, I was working on my own personal computer, so they would have access to every single personal thing that I had. And secondly, the software would send daily reports to my employer about every site that I've visited, and even remember passwords.
Can you imagine? It was suddenly the biggest invasion of privacy that I had ever experienced in my life. Some of my coworkers and I said no and asked if we could at least get work computers. Since they refused and the work environment was worse after that, I ended up quitting. There was no way that I was going to have that program watching me all day long.
So, to this Californian employee who's unsure what to do… I'd investigate the program they want you to use and see what features it has. In theory, what the company wants is to see when he gets to the client's home to do the scheduled repairs, but why would they want to know when you're at home? What are they hiding? Ask questions, investigate, and then make a decision.