Requirements!
You need to have maximum upgrades on the bugpedia, since it increases the chances of you catching shiny bugs, and the magnifier, so that you can recheck the same spot. Additionally, once you pass the 25th of August in the game, the chances of shiny bugs appearing increases. If you're at this point in the game, then you're good to go!

The main strategy that has been working for me:
#1 choose a specific bug that I want to find the shiny version of
#2 only hunt for bugs in the area/s where that bug is
#3 when I find the bug that I want, I only catch in that same spot so I'm increasing the chances of catching said bug again

I really wanted the spiny flower mantis, so earlier today I was only hunting in the crawler swamp. Once I found a regular spiny flower mantis, I clicked back on that spot repeatedly (there was a little golden star since it was a rare bug!) I did this again and again until I finally got it.

By only hunting for one specific bug repeatedly, I've had a lot of success! I hope that you can too.

Happy bug hunting!

In action!