I need to cut square holes in some double sided melamine particleboard 3/4 inch thick. Can I use a hollow chisel mortise (drill press style) to make these holes? I could just experiment with this myself but I will be using someone else's equipment to do this so I don't want to damage it. Thank you.
In addition to the particleboard itself, which is indeed quite hard on cutting edges (due to the glues binding the wood particles), the melamine coating is also very hard.
I would consider drilling as much of the waste as possible before squaring the holes, or at the very least drill through the melamine coating. Alternatively, depending how many holes you need, their dimensions and the accuracy required, I would consider doing it strictly by drilling the waste and squaring and cleaning with a regular hand chisel.
How clean do the edges of these holes need to be? Chipboard/Particle board can have a nice face surface but this inside could be a mess so I don't expect your standards would be too high here.
I would cut a groove into the melamine to help mitiagte chipping. Then use a drill, likely hand held, to make a hole large enough for me to get a jigsaw in there to remove as much waste. If appearances are not a factor I would be cleaning the inside edges with the jigsaw as well.
If you wanted cleaner edges I would pare with a chisel but like Graphus says:
[P]articleboard/chipboard is famously hard on cutting edges, quite astonishingly so sometimes.
And like I mentioned above I don't think you will get that clean surface inside the mortise but that might not be the case given the quality of the wood product.
In this Popular Woodworking article a chip free router bit seems to be a very good solution for cutting those through holes.
Spiral bits often make a smoother, more accurate cut than straight bits. An up-cut spiral bit pulls chips up and out, making it perfect for cutting mortises in solid wood. A down-cut bit pushes chips downward, ideal for making chip-free dados in plywood and melamine. (Down-cut bits are not recommended for use in a router table, however, because they can push the workpiece up off the table.) A compression bit has spirals running both ways, up and down, pulling chips toward the middle of the bit. It’s the best bit for routing the edges of plywood or melamine–you’ll get a chip-free surface on both sides.
You can drill out most of the material then clean it up with a router using this type of bit. If the corners need to be square then you get back to using chisels or possibly a hand held jig-saw to cleanly remove the last little bit of material.
I'm sure this can be done, but I think you're right to ask as particleboard/chipboard is famously hard on cutting edges, quite astonishingly so sometimes. So there is every chance the square chisels will need sharpening immediately after doing this, and it's even possible that a sharpening will be felt necessary once or more during the process if you're forming a fair number of mortises.