Based on your description, it sounds like the water inlet valve is making noise, since you hear a squeaking noise during the filling of the washer. If it sounds like a constant squeak or squealing noise, i'd say it's caused by either the water inlet valve not being pulled all the way open, or because the water supply shut off valve at the house end isn't open all the way. Try opening the water supply shut off valves all the way open, (check to see that they are).
What is interesting is that the squeaking comes back during the draining of the water. Is this while the tub is spinning or only at the very end of the cycle? Again, it could be the water inlet valve, since the washer will "spray" some fresh water into the washer during the final spin cycle to help remove any soap residue, etc.
The part ins't cheap at around $123.00 for this valve, so unless it's really annoying you to no end, I'd wait for the valve to stop working before getting the repair done, unless you want to do the repair yourself, which isn't difficult, with some basic tools and time.
If you wait and let this go until it fails, you'll notice the water in the tub might not fill all the way for the level set, or the valve will start getting even more as it gets worse.
Here's a drawing that shows your washer and the part I'm talking about.
Part #WH13X10017 $123.49 @ searspartsdirect.com
Hope you find this Very Helpful and best regards!