Warning: Anything you choose to do with your phone is your choice and at your own risk. This is merely of my own experience with the Droid x and the Droid Bionic. I'm not responsible if you fork your phone up. Follow this guide at your own risk!
So you're thinking that you do not want to upgrade your Bionic/Motorola device? Sick of the nag screens? Worried about the update slipping past and going through on it's own and not on your terms?
(See "Edit 10-27-2012" at the bottom of this post before continuing)
My method is to first root the phone and install busybox, there is no other way for this method.
Once successfully rooted you can grab esfile explorer which is free or if you want to support a developer, you can buy root explorer. Both will get the job done. Which ever file explorer you choose, you will have to allow it superuser privileges and you have to give the file explorer permission to read and write.
Now, go to the root of the phone which is "/", then go to "System/app and find BlurUpdater_VZW.apk and Upgrader.apk.
Rename the file extension to ".Bak" at the end of both files. They should now look like this. BlurUpdater_VZW.bak and Upgrader.bak.
I do it this way because when I set up a rooted phone, I set it up how I like, sign into all of my accounts and set all of my application settings to my desires, then I do a full back up with bionic bootstrap (Made only for bionics).
If you find this guide, I imagine it should work on all Motorolas the same way. But keep in mind, each Motorola device may have it's own nandroid and bootstrap app and this guide was designed for Verizon in mind, I don't know if the files are different for other carriers. I do it this way because if I have to restore a backup, I already know those files will remain named they way I chose in case I forget.
Or, I believe you can simply just freeze them. I personally don't care for apps that freeze, I like to rename files, but that's my personal preference. I'm not a professional at hacking phones. If you have tips and different methods, please share them.
I'm writing this guide because when I wanted to know how, I spent hours looking. So I deliberately titled this post to show up on the search engines.
A special thanks to doogald for helping on this particular issue.
If you by chance downloaded the ICS upgrade and do not want to upgrade to ICS. Before rooting or renaming files you should clear the cache folder (Where the download is believed to be).
1.) Shut the phone down.
2.) Hold down both volume buttons at the same time (Vol-up/Vol-down). Then press and hold the power button. This will get you into stock recovery.
3.) Press the volume down button one time to highlight recovery then press the volume up key one time. (Pressing on the volume up key is how recovery is selected to execute)
4.) Wait a moment till you see either a triangle (Gingerbread) or an android figure with his middle section opened (Ice cream sandwich).
5.) Once a again, press both the volume up and the volume down keys at the same time. This will bring you into the desired recovery area.
6.) Use the volume down button to highlight "Wipe cache partition".
7.) Press the power button and the cache will be erased/formatted.
8.) After format is completed, use the volume up button to highlight "Reboot system" and press the power button.
9.) Follow above section to then rename the files.
I would guess to say it would be better to wipe the cache first so the update doesn't start accidentally on a reboot while rooting. I don't know that it would, I felt this would be this best option to format that cache first and rid the system of the update files first.