There's quite a bit of advice on the web about how to fix squealing hydraulic brakes. I have put together the good advice I've seen, and my own observations and experiences.
Please add your own ( and, also, let the flaming begin)....
If your hydraulic brake is squeaking or howling, I’ve found, through making many mistakes, that the following helps…
1. The awful sounds are usually caused by oil contamination of either the rotor or the brake pads, usually both (they contaminate each other). Note: the brake pad can absorb oils.
2. The brake caliper often has a buildup of crud on it (as does the surface of the pads, especially if the pads have cooling fins, which can trap lots of crud). When you remove brake pads, this can disturb the crud. Some of this often oily crud may drop onto the rotor or brake pads. Removing the wheel from the bike is best to prevent oily crud from hitting the rotor when removing the brake pads.
3. To clean the rotor (don’t touch the surface of the rotor with your hands while doing this… your natural oils can contaminate the rotor):a) Removing the rotor from the wheel makes the following easier and probably more effective.
b) Wash the rotor with soap and water to remove water-soluble contaminants, then thoroughly rinse the rotor, then dry using a very clean cloth or paper towel.
c) Using steel wool and rubbing alcohol (or bicycle-specific disk brake cleaner) to remove oils from the rotor. Aggressively clean both sides of the rotor using the steel wool.
4. To deal with the brake pads:a) Remove the wheel to prevent contamination of the rotor that could be caused by crud dropping from the caliper onto the rotor when you’re removing the pads.
b) Remove the brake pads.
c) Thoroughly clean the entire surface of the caliper, including the surface of the pistons, and the areas surrounding the caliper using rubbing alcohol or a bike-specific brake cleaner, applied to a clean cloth or paper towel. The brake dust and other dirt that gathers around the caliper can absorb oils that your bike picks up during your rides. Check for any buildup of crud around where the hydraulic brake hose enters the caliper… this indicates that you may have the tiniest hydraulic leak, which will probably cause contamination. If you have a leak at the caliper/hose connection, it’s doubtful that tightening the hose connection will help… for this connection, tighter doesn’t mean better. If you haven’t had experience with brake hoses, take the bike to a bike shop and tell them about the leak. They’ll separate the hose from the caliper, replace the hose insert and olive, and tighten the hose back onto the caliper to the correct torque.
d) If your pads are very lightly contaminated, try rubbing the surface of the pad against waterproof sandpaper, laid flat on a table. Do so in a figure-of-eight pattern. A few drops of rubbing alcohol or brake cleaner help. I’ve found that cleaning the pads is rarely effective, so I now just buy new pads, making sure that, before installing them, the wheel is off the bike, the caliper body and the rotor have been thoroughly cleaned, and that there isn’t a leak from the junction of brake hose and caliper.
Some suggest baking the pads in an oven, although I haven’t tried that yet.
One thing I learned the hard way is that the rotor and pads contaminate each other. Cleaning, or replacing, one means that one must clean, or replace, the other.
5. When bleeding the brakes, I find that I get oil, which can contaminate rotors and pads, everywhere. I now remove the wheel (remember to put a caliper spacer into the caliper when you do this; it’ll prevent the pistons from moving too far towards each other if you accidentally press the brake lever) and brake pads and put them in another room before starting the bleed process. After completing the bleed, clean the caliper and area around it using rubbing alcohol or a bike-specific brake cleaner, before replacing the wheel and brake pads.
6. After cleaning/replacing the rotors and pads, bed-in your brakes.
7. Caliper alignment may also be a problem, but that has not been a problem for me, therefore I won't add my two cents to advice already on the web.
My system is Shimano Ultegra 11-speed hydraulic Di2.