I will be moving to Switzerland in July and my company is already applying for my work permit. The question I have is that my partner (female) and I already have a 2 year old son and she is pregnant with our second child that is due in May. Must we get married in order for Switzerland to grant her and my children a living permit in Switzerland with me, or can we stay unmarried as both children are mine and they still receive a permit to live with me in Switzerland?

firstly, I had to investigate you seem to be a US citizen with a DUI (nothing to worry about).

you should ask your employer to get permits for you all and relocate together. Marriage would make it easier (but maybe harder too )

We are all US citizens and for personal beliefs we would prefer to stay unmarried but I was wondering if we must do it for them to get the visa? My company will apply for them to get this visa as well, and our plan is to move at the same time together to Switzerland in June.

I don't think you have to be legally married but would need proof of cohabitation and the children recognised as being yours. I'm sure when your company insure your permit is available they will have no problem with that of your family. They will get advice on the paperwork needed.
Good luck with the move! And new arrival of course!