Installing a new rocker style switch and the junction box had one two black wires connected to the existing switch (hot and neutral)
In the box there’s two copper wires braided together and two white wires twisted into a floret
Should I connect the two braided copper wires to the ground on the new switch?
First of all, the two black wires are not "hot and neutral", they are "hot and switched hot", or alternatively referred to as "line and load". Neutral is always white (well, occasionally gray), but black is never neutral unless there is something wired incorrectly.
A standard ("dumb") switch connects to hot and switched hot. It doesn't connect to neutral. It also doesn't matter which wire is which, because the switch doesn't do anything except connect/disconnect the two wires.
On the other hand, a "smart" switch, dimmer, WiFi, timer, motion sensor, etc. will often require neutral. You are in luck if that is the case as you can simply add another short white wire to the existing wire nut containing two white neutral wires (you may need a larger wire nut). That is not always the case - many older switches do not have neutral in the switch box, but you do. A "smart" switch/etc. may also be particular about hot vs. switched hot, in which case you would need to figure out which black wire is which.
As far as ground, the ground wire needs to connect to a screw in the box, and it looks like it does. You do not need to connect the ground wire to to the switch when using a nice metal box like you have. With receptacles you sometimes need to connect ground to the receptacle, but with better quality ("self grounding") receptacles you don't. This is a nice benefit of using metal boxes.
