Next, take the fuel line hose clamp and clamp the brake line near the flare fitting. Others have suggested using vise clamps with rubber hose on the ends and this may work, but I don't have any extra rubber hosing that fits my vise clamp, so for $8 at pep boys, I picked up the correct tool. No doubt I will get to use this again in the future.
See attached picture. Remember, this clamp is only to keep brake fluid from completely draining out of the hose. It will still leak a little.
Now, while the caliper bolts (or bracket bolts) are loose but not removed, take your 14mm socket and 3/8" drive wrench and slowly undo the banjo bolt. Brake fluid will start leaking out. Because of the orientation of the brake line opening in the caliper, the fluid in the caliper will pour out on you when the banjo bolt is removed.
Place the banjo bolt in a safe place, if reusing, after spraying it down with brake cleaner. Gently move the brake line out of the way for full removal of the caliper (with or without bracket). Remove the bottom bolt on the caliper (or bracket) first. Be ready to catch the caliper as you remove the last bolt!
I have been told by several mechanics that it is perfectly OK to reuse the old crush washer if it looks good. Since the washers on my 2003 AWD RX are the new style steel washers, I tend to agree. See the first image in post #47
to compare with what you find. According to Toyota's service manual, 28 ft. lbs. are required to put the bolts back in place. I think I went a bit more than that just by hand when doing the 1/4 turn past snug.
Now, once the caliper has been swapped, follow this tutorial in reverse order for reassembly. Once all bolts are back in place, proceed to bleeding the brake line.