Interesting. When I hear June Bug I think of a large green beetle I grew up with. We'd tie a thread to their leg and fly them. I read this "Green June Bug", Cotinis nitida, is restricted to the Southeast where that brown one is all across the country. I'd never heard of a brown june bug. You learn something every day.
The Green ones are out during the day and will chew up certain fruit. They swarm to ripe blackberries. Last year I noticed the Green ones would swarm to an apple a bird had pecked and eat the whole thing. The green ones were pretty bad here the last two summers.
I have no idea on the habits of that brown June bug.
My dogs eat June bugs. We have a security light that comes on at night, it draws June bugs. During the day, the bugs burrow into the sand, the dogs dig them up for snacks. Last summer, they ate so many, their poop was assorted indigestible bug parts.
I wonder, are they attracted to light? If so maybe you could hang a bulb over a basin of soapy water at night and catch some that way.
It works for fleas, haha.
Yuck! It won't help this year, but there is are grub killers for the lawn that may take care of them in future years. I'd not use it on the garden soil, but on the lawn I might.