Are the Zerg fully biological beings or are they bio-technological as described here, like the Protoss who are half machine and half Protoss?
The Zerg have similarities with the Tyranids of Warhammer 40k, these are organic organisms themselves, but can be defined as biotechnological since biotechnology are defined with this:
Biotechnology is the use of living systems and organisms to develop or make products, or "any technological application that uses biological systems, living organisms, or derivatives thereof, to make or modify products or processes for specific use"
The zergs use other species to improve their "swarm", this process is carried out through modifications of their DNA in their larvae, this generates the different "species" or forms of Zergs that exist. Therefore we can define that if they are users of biotechnology and can apply it in their own bodies.
Another very different thing is what is said in one of the comments about the "infection", since the processes are completely different and have no relation whatsoever.
Protoss are not "half machine", they are simply a race with very advanced technology, in addition to having certain psionic abilities. But in no case are they a half-breed robot or something, use armor and weapons just like the Terrans use them, but they are not half machines.
The Zerg are biological by nature. They do have the ability to control alien technology by infesting buildings and beings (such as infested Terrans and Alexei Stukov, who controls many different "bio-technological" infestations), resulting in biological machinery, so to speak.
The Protoss utilize cybernetic implants to enhance their psionic abilities (and to teleport away when nearing death), so in a way you're right in that they're half machine and half Protoss.