Brake light do not i check all lights and fuses are good.

Change the bulbs.

Ok, trailer lights are basically separate circuits from the regular stop lights and same with the center brake light.

There is a brake light fuse located in the fusebox next to the battery in the engine compartment. Use a 12v test light to see if it becomes hot both sides of the fuse, when the brake is depressed.

12v Test lights can be bought at a auto parts store for less than $15. Do not get a "continuity light" as these are different. A test light, lights when you touch a terminal and voltage is present using the alligator clip attached to a good ground

Notice that the center stop light does not go through this fuse in the diagram.

I have included a wiring diagram for your vehicle.







If you find the brake light fuse blown, and it repeatedly blows, you have a short to ground from the fuse to the brake lights somewhere. First step is to check the brake lights for a short. and work your way back to the fuse.

Since the turns work in the back, that means the brake bulbs are good. Brake lights and turn signal lights share the same filament in the bulb.
There is a fuse for the brake lights. Then, there is a brake light switch on the brake pedal. You need a voltmeter or test light, after checking that fuse is good. Need to check if fuse has power on it. This is one of those fuses that are hot at all times-as long as battery is connected, voltage should be going through the fuse and up to the brake light switch. Check for power both places. The switch on the pedal is a simple on-off switch. When brake is depressed, the spring loaded switch is turned on-passing power on towards the brake lights.
But then the brake wire goes into the turn signal switch before going on to the brake bulbs...however, since your turns work good, I doubt if there is a problem in the turn switch. First check if the brake light switch is sending power on when switch is on.
Let me know what you find. Good luck.

Buy a cheap test light to run simple tests to see where power is being lost. Start at the brake light fuse. Clip the ground lead of tester to a metal bracket or bolt under the dash. Then use the tester's probe to touch the little indents on top of the fuse. There are two indents on each side of tops of fuses-these are test ports to check if a fuse has blown. The brake light fuse will be "hot at all times", so touching the metal strip inside the indent should light up the test light. If not, check if your ground clip is good: turn the key to on. Now every fuse in the dash panel should be powered. Every fuse should light up the tester. If both indents on a fuse light up the test light, that means it is a good fuse. If only one side lights up the tester, that is a blown fuse.
Now back to the brake light fuse, key off. Does it show power? If not, maybe a fuse link off the battery that distributes power to the fuse panel has burned out. If fuse shows power at both ends, move tester to the brake light switch and check for power at the switch-this wire to brake switch is again "hot at all times". Should have power in, and when brake pedal is depressed switch should also have power out. From the brake switch, the brake signal wire goes up into the turn signal switch, and is then routed out to the brake lights. With pedal depressed, you can check for power in and power out at the turn signal connector. Don't pull the connector apart, but back probe the connector with the test light probe. You know the color of the brake light wire from brake switch to the turn connector, so easy to check for power in. Power out should be on two wires, one for each side of the rear lights. If the turn signal switch has power in but no power out, with brake pedal depressed, probably a defective turn switch.
Post back with results, or when you find the problem.
Good luck.

Check the brake fuse in your dash's fusebox. Since the turns work, you know the bulbs are good, unless your Neon has separate brake lights. If the fuse is good, bulbs are good, then check for power at the brake light switch on the upper portion of the brake pedal. Should have 12 volts there all the time. Use a testlight or voltmeter to check. Good luck.

The third or High Mounted Brake Light runs straight off the brake light switch on the brake pedal arm. You say you tested the "brake unit"? I assume you mean the third brake light assembly, and if you get power there when someone steps on the brake pedal, then the bulbs are bad.

The power flow for the brake lights goes like this: Fuse ->Brake switch -> Turn signal switch -> Rear lights. The only difference is, if you have a 3rd brake light in your rear window, that gets its power directly from the brake switch. It does not go thru the signal switch first. So, if you are having a brake light problem, and your fuses are good, check the 3rd light in the rear window, if it lights up (assuming the bulbs are good, no one ever changes them), then the problem is most likely in the turn signal switch, hope this helps!

to clear the code disconnect the battery, you could have a bad brake lite swich, its under the dash by the brake pedal

WITH A GOOD FUSE TESTER, TEST ALL THE FUSES FUSEBOX UNDER THE HOOD & INSIDE THE VEHICLE. IF THE FUSES ARE ALL GOOD WITH THE TESTER YOU MUST HAVE A BAD BRAKE LIGHT SWITCH OR A BAD TURN SIGNAL SWITCH. THE BRAKE LIGHT CIRCUIT GOES UP THROUGH THE TURN SIGNAL SWITCH & THEN TO THE BRAKE LIGHTS. FIRST OF ALL IS THE BRAKE LIGHT SWITCH GOOD ? THE ORANGE WIRE FEEDS THE BRAKE LIGHT SWITCH, THE WHITE WIRE PUTS POWER TO THE TURN SIGNAL SWITCH, SO THE LIGHTS CAN BE USED AS TURN LIGHTS OR BRAKE LIGHTS. , PUT A TESTER IN THE WHITE WIRE & PRESS ON THE BRAKE PEDAL. IF THE TEST LIGHT LIGHTS UP, BRAKE LIGHT SWITCH IS GOOD, BY THE WAY THE BRAKE LIGHT FUSE MAY BE LABELED" STOP", THE FUSE LABELED" BRAKE" IS FOR THE ABS SYSTEM. TO CHECK THE TURN SIGNAL SWITCH FOR BRAKE LIGHTS, REMOVE THE PLASTIC COVER FROM UNDER THE STEERING WHEEL, PRESS ON THE BRAKE PEDAL, PUT THE TESTER IN THE SMALL YELLOW WIRE, THEN THE SMALL GREEN WIRE. IF THERE IS NO POWER IN EITHER WIRE, THE TURN SIGNAL SWITCH IS BAD.

If the two rear brake light are working on each side, then fuse and relay are good, check the bulb.

check the bulbs, check the connector ,check ground and voltage power check fuses and check brake light switch right above the brake padal.