You can do whatever you like; the company can also do whatever they like. Unless it's stated in the contract that you don't have to use your own personal equipment, or it's in some law in the Philippines (it probably isn't, as I'm unaware of any countries which have such a law, although I'm unfamiliar with Philippine law so perhaps it is; you may want to research this), the company can request that you install software on whatever machine you are using for work. You can decline the request, but the company can fire you for that (even if they can't fire you directly for that, they can make up some excuse to fire you based on that, like "you aren't being productive enough" or whatever).
So really it depends on how much you want this job. The company, if they are a good company, should provide you with tech; there is no guarantee, especially in a third-world country like the Philippines, that every employee has the equipment at home necessary to do their job. If you need to run some kind of CPU-intensive operation, or you need a high speed internet connection, not everyone can do that. Furthermore, it's in the company's best interests to not do that, because if your personal device gets hacked or whatever then you could leak company IP, and it's not reasonable for the company to legally hold you to that standard.
If you want to keep this job, then install the software. Not that it's ethical for them to make you do so, or that you should, but simply put if you don't install it then they'll probably fire you, so it's a choice between using the software or losing your job and you said you don't want to lose your job. However, you should notify, in writing, the company and say that, because you are being forced to use your personal equipment for work, you cannot guarantee the safety or security of any company IP that is on your computer, and make the company agree to that in writing, so they can't hold you legally liable for this later just in case you happen to get a virus or what have you. If they won't agree to this, then I'd say you should just find another job because this company sucks.
The other option, if it's reasonable for you, is to buy a second computer to use for work, and install the software and all your other work stuff on that computer. You should still get them to agree to the terms of you not being liable for the company IP and so on but it's less of an issue if you know your work computer is more isolated from your home life. However, a new computer can be expensive and it may not be financially reasonable for you to do this, in which case, see above.