I looked and found the exact BIOS your system uses, from what I can tell they do not have an option to boot from USB (even though that HP employee said they do). You could get another tool, download it, boot it from a different disk, which allows you to boot to USB, but I would like to find a simpler solution.
I am guessing you are probably more comfortable with Windows environment, and I hate to direct you to a tool I haven't used, but you could try doing all your partitioning and mounting of the disk image (burning the .iso onto a drive letter). You could try creating a partition in Windows, then use OSFMount and rename the .iso to .img, then mount it to your new partition.
I want to stress I am not sure whether this will work, and I would highly recommend if you want to try this way, back up anything you need, write down your reg key for Windows, and make sure you have a disk.
Before doing the stuff above.. I see a couple ways of going.
You go through the partition, and mounting above, lots of backing up of information, and not guaranteed to work.
(The one I would suggest) Make a LiveCD of an older version of Ubuntu like 12.04 LTS (5 year support, ends in 2017), then if you want update it to 14.04. I am only suggesting this because 14.04 JUST came out (14 is for the year 2014, April 04). There will be bugs, just like any OS release. It will be a lot easier if you just get a working version, then if need be upgrade from there.
If you are in love with 14.04 and want to fix it we can go that route, but if you do wish to go that route, I have another question. What do you have in advanced options, when you select it from GRUB. I am used to having a recovery option, but I do not dual boot Windows with my personal computer, but have set up quite a few XP,7,and 8.1. If you have a recovery option, that may be the easiest way to get Ubuntu 14.04 up, if your computer will actually run it.
The only reason I wanted to run a live version, is so that we could find out what might be the problem (almost like running Windows in safe mode).
Older stuff below:
I do not believe that the first Ubuntu install is correct. Normally it would show a version behind it. I have a strong hunch that the filesquash.fs error is because of the loader trying to extract itself onto itself.. that sounds weird, but I believe that is what the message is saying.
If you have no clue what I mean: Basically to me it seems for some reason the LiveCD is trying to install on top of itself. I am not sure what the issue is exactly, just save the disk there may be no problem with it, but if you have a spare, you can always try a different disk. I know you followed the instructions by using a native Windows 7 program, but I am going to direct you to a different way of using a bootable USB. (Less expense if you mess it up accidentally, just redo, not use a new disk.)
How to create a bootable USB stick on Windows.
I looked up your motherboard, because I wanted to double-check that you could not boot from USB (it just seemed odd because most newer Motherboards let you boot from a USB). It looks like you can as per an HP employee, you just need to make sure that you have a bootable USB in your drive when booting up.
(Mobo Spec Link) Just in case you wanted to check too.
Use a spare flash drive, if you have one. It is 2GB minimum for the bootable USB but I chose to use a higher capacity and speed one (USB 3.0), because that's how I roll.
If you can get this to work (hopefully), you are going to need to know how to install/partition and mount. You can look through the Ubuntu installation guide to figure it out, just know that it may be easier for you to partition a drive in Windows (unformatted) prior to installing Ubuntu.
The reasons why:
Ubuntu could overwrite something (probably not though unless you install over top)
Windows could get mad at Ubuntu, and try to 'fix' itself.
It is cleaner, and easier if you are used to Windows.
You could partition incorrectly with Gparted if you don't know what you are doing.
I need to make food for my wife right now. I will come back later to see if there has been any progress and if this has been helpful.
Yes I am willing to help. I am doing this outside of the comments, so that there is not a conversation below your normal post.
I am going to try and figure out the whole scope, so I am going to ask a couple questions. I know you were able to get Ubuntu installed before, but currently what is the situation? I am not trying treat you like you are stupid, just want to follow your steps one by one.
What brand of computer is this? Are you able to get to your BIOS/UEFI?
Did the installation process go through, or did it fail at the filesystem.squashfs error? (Like did you to the part where you partitioned your drives, set a mount point '/', or '/home' or something like that?)
Does the error occurred before the installation (the logo will come up before you can liveboot)?
If you get to the 'Try Ubuntu'/'Install Ubuntu' Screen are you going straight to Install, or to Try Ubuntu?
Hopefully we can get to the bottom of this. If you could add this information to your question that might help.