Automatic and when put into gear (reverse or drive) the backend starts to grind. Roll backwards or forwards the car feels like the parking brake is on and the front end starts to squeak.
What could this be?
Sounds more like the differential ot tranny. Have you tried getting it rolling and then put it into neutral? Do the symptoms go away w/ it in neutral?
Since I can’t hear the sound I’ll make a suggestion you can test easily. If you have drum brakes on the back, you could be hearing loosely adjusted brakes. It would be caused by rust hitting the brakes. Go to a parking lot that’s empty and back up while repeatedly pumping your brakes to see if they self adjust up (that’s how drum brakes normally adjust themselves). If you don’t do a lot of backing up you can get into a loose brake situation. You could also be dragging a brake because the parking brake cable has rusted. Had that happen to an 87 thunderbird. Although it had disc brakes in the back the principle is the same.
If it’s a really loud grinding noise it is possible for the axle to pop out and grind on the shaft (happened to my dad’s 67 olds) but it’s highly unlikely. I’ve never heard of another vehicle doing it.
It could be many things. Best way to diagnose this is with some jackstands. Get the front in the air and turn each wheel by hand, feeling and listening for bad sounds. You could have a bad wheel bearing(s), a hung caliper, or disc pads that have worn to the point the backer plate it rubbing the rotor. Assuming the front passes, get the rear on jackstands and do the same thing. Check the differential for evidence of leakage, and pull the brake drums to see if the outer seals are shot, allowing leakage there. Pop the differential oil plug and see if fluid is up to the level of the plug. If you note a noise in the rear axle, but can’t tell for sure where it is, have an assistant rotate a wheel while you use a length of heater hose held to your ear and pressed against various parts. The hose will act like a stethoscope and help to isolate the noise. With the vehicle safely elevated and the trans in neutral grab the trans output yoke with one hand and the driveshaft with the other. Feel for any looseness between the two, as evidence of a bad u-joint. Repeat on the other, or other 2 u-joints.
What gets me is that the sound in the back occurs without the truck moving. Simply shifting into reverse or drive does it.
Noise upon shifting into gear while stopped can be a number of things.
Bad U joint
Bad motor mount
bad trans mount
bad bushing(s) on rear axle
clearance between brake pads and caliper (Loud click)
Transmission issue
I would note that all of these are usually a clunk or a thud or a click. A grinding noise while stopped has to be in the engine or trans. as the driveshaft and rear axle are not moving while stopped. A bad motor mount allowing the engine to shift when put into gear, for example, can cause such a noise.
The squeak in the front end is not related I am sure. Usually a squeak in the front end is bad/dry bushings or a lubed joint that has not been lubed.
Drain the differential into a container and look for small bits of metal.
If the truck is not moving, the diff is not moving. If the diff isn’t moving, it isn’t making any noise. Draining the diff won’t help locate the noise.
Yeah, I missed the part where the noise existed w/o the trck moving. First step would be to put the rear wheels off the ground on jackstand and then get under there and check things out, wiggle the drive shaft, put the tranny in neutral and turn the rear wheels by hand, put the tranny in gear and listen for the location of the noise, etc. I’m no great mech., but that’s where I’d start. I’ll bet you can get a much better idea of what the problem is by doing that. Just make sure the vehicle is stable and wheel block before you crawl under there. Hard to diagnose just by sound, you need to get your hands dirty, up close and personal.
Before you break out the jack stands, I would just get someone to sit in the driver’s seat, and start the engine. With their foot on the brake, put it in gear. Walk around the car and see if you can locate the noise. If it appears to be under the hood, put on the e-brake and open the hood and look.
You might get way lucky and find it without jacking the car up.
Give the hard job to the lazy guy, he will find the easy way to do it. 
You can’t have a grinding noise from the rear of the truck if it isn’t moving. You could have a rattle or vibration from parts touching. If you’re getting noise from the back when it is in gear but not moving then the exhaust probably is shifting from the torque of the motor. I agree with Rick. Have someone sit in the truck with their foot on the brake and put it in gear while you walk around looking for the noise.
Update
I finally got a chance to drive the car around the block
Brakes grind. My stepson is very hard on brake and never inspects them until they grind.
It feels and sounds like the brakes are grabbing when going in reverse. Problem ranges from annoying to really bad.
Absolutely no problems going forward anymore - even at freeway speeds
An occasional slight shimmy when coming to a complete stop. Very intermittant.
So is it the brakes (my first instinct was brakes or u-joint)? If so, what could it be that only affects it going backwards? If it is the rearend being in reverse, then why would it affect it before in forward but not now?
Assuming this is a ‘when in motion’ noise:
If your truck has drum rear brakes, the adjusters are trying to do their job in reverse. It sounds as though you need to get the wheels and drums off to see if damage is being done, because if you tear up the drums too badly and they can’t be turned, replacement is the only option.
Nope. Disk brakes.
But I see there are a lot of issues with the Ebrake sticking. If this is the problem how do I unstick it?
Ford trucks (not sure if all) with disc brakes on the rear have a pair of small brake shoes which press against the inner hub of the rear rotor to apply the parking brakes. Like any parking brake, cables can sieze, or there could be another problem with the parking brakes (broken spring-worn shoes). Siezed cables should be replaced.
Parking brakes are often ignored until they cause problems. Last week, I redid brakes on my E250 (four wheel disc), and took the time to lube the backer plate, the adjuster, and verify that both sides were applying and releasing properly.
IMO, most parking brake sticking problems result from lack of use. When the cables aren’t exercised, they stick.
So how do I unstick it if that is the problem?
Replace. The. Cable.
What he said.
If you are not familiar with doing drum brakes, send it to a shop. Trust me on this.
Aye. Ford particularly has the nasty habit of adopting a technology which is pure shite, using it for a model year (or less) and then dumping it in favor of something better (thus making it difficult to find parts/fix if you’re the poor bastid who has that particular set up). I had an '88 Lincoln Continetal with a screwy parking brake set up that caused myself, my brother (an ex-mechanic) and his best friend (a helicopter mechanic for the military) a couple of hours of pain while we tried to get the spring back on one of the calipers (bizarrely enough, the caliper on the opposite side of the car was a breeze to deal with, something about the handedness of the parts, I guess).
That would be the easiest fix. I’ve pressurized aircraft cables before and forced oil though them but it would be tough to do with a brake cable.
Saint Cad, are you sure this is a disk brake truck? All the Ford 150’s I’ve looked at were disk up front and drum in the rear. The reason I ask is because a drum brake will often grab backing up when it’s too loose. The grabbing sound would be multiple and clunky.