“The high-pitched brake squeak is likely from the brake pad wear indicators telling you that soon you will need new brake pads on your four wheels…”
In my experience, the brake pad wear sensors make noise when driving without pressing the brake pedal, and the noise tends to go away when the brake pedal is pressed. What the OP describes appears to be the reverse of my experience, so…
In any event, the brakes need to be examined by a competent mechanic in order to determine exactly what is going on.
As to a brake fluid flush, unless this service has been done in the past 3 years, then–yes, it is due for this service. IIRC, the Honda maintenance schedule does call for this service every 3 yrs/30k miles. Has the OP bothered to take a look at the mfr’s maintenance schedule that should be sitting in the glove box?
As to flushing the power steering, that is something that is a judgment call, IMHO. I do it on the same schedule as brake fluid flushes, but I tend to keep my cars for a long time. If the OP is planning on getting rid of this 9 year old car soon, then he might want to skip the PS flush. However, if he expects to drive it for several more years, I think that it is a good idea to get new fluid, with new additives, into the PS system.