I used Universal USB Installer v1.8.7.3 to install several Ubuntu variants to USB flash drives.
When you boot from the USB drive, with whatever "*buntu" flavor, there should be an icon on the desktop labelled "Install *buntu".
That's how I did the Xubuntu install to my SSD drive.
And it worked. Smooth sailing, except for the space constraint.

Universal USB Installer provides a "persistence" option that UNebootin doesn't seem to offer. That lets you save changes to the Live environment, like retaining installed packages, across reboots.
Of course, the persistent casper-rw filesystem tends to self destruct before too long, if you don't manually run fsck on it from time to time... (see my previous post)

Since the Live distribution on the USB drive seems to get by fine without a swap partition, you can probably reduce the size of the swap partition that the OS installs and recover some space that way.

I don't know how you're using the SD card right now (and I don't know how flexible the installer will be) but you might consider getting a fast SD card ("class 10", "133x", or better) and making it a permanent part of the system by putting the /usr filesystem on the card, so that packages and libs install there, instead of on the internal SSD.

There are several netbook and eeePC-specific distros out there (Lubuntu, Peppermint OS, Ubuntu-eee/Easy Peasy, Leeenux, Eeebuntu/Aurora OS, etc.). I'm really not clear on the differences. All are supported by Universal USB Installer...except for Aurora, which has its own installer.

Post back and let us know how the Lubuntu alternative ISO works out on your eeePC.