Can I declare I won't be using their app; what rights do I have?
It's a personal property and you are an employee, which means that the employer cannot force you to install anything on it. They can act nicely, but there isn't really a path (as far as I know) which can results in you being forced to install software for your employer on your personal phone.
But as importantly going down to legalities is almost always a losing route.
First and most obvious they can then not extend your work contract/dismiss you at first opportunity (and with bigger companies and very thick contracts and handbooks there will always be a way at some point), and less obvious they may start enforcing rules that you may not want enforced. For example it's not uncommon term in retails that employees are not allowed to use personal phones during working hours. While that term is seldom enforced there is no reason that this cannot change.
but lets assume I don't have a smart phone anyway, then what?
You need to think carefully about what you want to achieve before doing anything.
I definitely would not start by lying about not having a smartphone, this will be seen through and found out almost instantly bringing the topic back to the original question but now your credibility is also shot by this obvious lie.
If you do not want to stir the pot and keep everyone happy while also protecting your own privacy then the easiest option may just be to buy a cheap 2nd hand phone. While this will cost you some money out of pocket it avoids rocking the boat and possibly (although unlikely) putting your job at risk. You can even leave this work phone at work assuming that you have some sort of storage options available, so even if you will forget to turn off the tracking it won't track you outside of work.
If you want to challenge this change then you really can only hope for two outcomes:
- You getting assigned a work phone from the company. Given that they just do not do that it seems unlikely to happen but you won't know unless you ask. You can possibly increase your chances by having a proper pitch about how a work phone would help you, when not having to mix work apps with your personal phone is just one of the many great points, though that really depends on your relation with your boss.
- You refuse to install the app and hold your ground. They will then either figure out an entire new procedure to accommodate you (which they may already have, who knows, though expect it to be relatively paperwork heavy) or they will smile, say "okay" and start working on a way to replace you because who needs that pain. And there are many ways to fire an employee, or drive them into quitting if your boss cares enough.
Whichever route you want to take we cannot tell you, but I urge you to think it through before acting, as a label of someone who makes a final stand on what most people consider a non-issue (and many people just do not care about being tracked, especially when it has an off button) is a pretty hard one to shake off.