June is pretty limited: most big game hunting is done in the fall. In June you may find a spring bear season. They are controlled by the individual state as far as the seasons. a "canned" hunt- a hunt with an outfitter with pen raised animals, behind high fences- on other words, not wild animals, is another option. They are not controlled by the state, the game being non- native or domesticaly raised. A fair bit of the south (and possibly out west, I am not sure) have year round wild hog hunts. They are considered vermin, and can be hunted year round. It should be easy to find a guide for them, depending on where you want to go.
Where are you going to be, or are you planning to come to the States primarily for your hunt? I believe Maine has spring bear hunts, and a google search for spring bear hunting would tell you more. Just over the border of Maine is New Brunswick, which has spring bear hunts also, with many good outfitters. Rifle renting is generally not done, but when you call outfitters and guides, tell them your situation: many would be willing to loan you a rifle. At least in New York (where I am) a non-resident may hunt without a guide, but you need proof of a hunting safety course where you are from to get a license. Of course then the tough part is figuring out where to go: NY has millions of acres of public accessible land, but not all of it is appropriate for hunting, and none of it has open seasons until fall. Good luck, you have alot of options, just limited quarry in June.