Electrical – EAGLE Square hole

No, you don't want just a oversized via. Step back and think about the problem a bit. You don't just want a mounting hole, but rather a genuine electrical part. This part should show up in the schematic, be something you place on the board, have a pin connected to a net, and a pad that connection can be routed to, just like other electrical parts.

The correct answer is therefore to use such a part. This would have a schematic symbol with one connection. The package would be a thru-hole pad with the hole size matching your mounting screw and the pad diameter a little larger than the screw head diameter. If you want this to be a ground connection, connect it in the schematic to your ground net.

I have done this before and have a few such mounting hole parts in my library, including one for a #4-40 machine screw.

Use a via to a track and set via pad size to whatever you wish.
There will be a minimum size to obtain reliability.
Your PCB supplier will (or should) have design rules covering this.