westbay,
Big Bend is a great place to ride if you have a street legal bike. There is no public land there that I know of that you can ride on a non-street legal bike. Everything off-road out there is on private land.
Big Bend is very large and can be very remote, depending on the route you select. The paved roads will have some other vehicles on them, but the unpaved roads will have very little traffic. I road the 50 mile unpaved river road in Big Bend park this past Christmas and only encountered 2 vehicles the entire day (and Christmas is one of the busiest times of the year at Big Bend). Decisions about what to carry (gas, tools, water, etc.) should be based on where you ride. On unpaved roads I'd operate of the assumption that if you break down out there that you will have to walk out. On paved roads, eventually someone will come along, but it may be hours in the hot sun till then.
Mid-June will be very, very hot - likely 100 degrees every day with 0% cloud cover. You will want to wear lots of sunscreen and make sure you carry lots of water with you, especially on the most remote of roads.
You won't need anything special in the way of protective gear. Your normal armored jacket & pants, gloves, sturdy boots, and helmet will be sufficient. I do recommend vented clothing though. If you are going to ride the unpaved roads in the area, knee pads are a nice addition.
There is a cycle repair guy in Terlingua Ranch, but if you break down in the boonies, you will need sufficient tools to repair your bike or a tow line so you can tow your bike out (assuming you aren't riding by yourself).
Finally, there aren't all that many road choices in the Big Bend area. The roads are very, very long, but there aren't a lot of different roads. Use the search function on this forum and read all the Big Bend ride reports. They will give you a good idea of which roads are the most fun.
Have a great time.