So I was recently recording with my Jamman stereo looper. The recordings came out tiny-sounding because it is just recording the direct guitar signal. I was thinking of plugging the input of the jamman into an external speaker jack in my amp so I can record the signal after it passes through the tubes and what not for a better sound. Is it possible to do that? Can I potentially damage my amp or looper if I do this?
Yes, you will absolutely 100% guaranteed ruin the jamman doing that. The output of a guitar is somewhere in the range of a few milliwatts (1/1000th watt) of power, the output of even the smallest of tube amps are 0.5 - 1 watt, and more commonly 15 - 120 watts. Totaling to between 500 and 120 000 times what the jamman is designed to handle. Also the impedance of a speaker is 2 - 32 ohms, whereas the average input impedance of a pedal like the jamman is ~10 kilohms (10 000 ohms).

If you have a headphone out and use that you shouldn't damage it, but make sure you still have speakers plugged into the speaker out or you'll damage the tubes. You could also use the FX Send if you have an effects loop.