seized bottom bracket
 

I put a square taper bottom bracket in my bike and it kept coming loose. The vendor told me to wrap the threads with teflon tape approximately 6-7 times. I reinstalled the bottom bracket which was a little tough getting on (big mistake). Bottom line: its seized and the park tool is stripping out the aluminum cups. How might I get this thing out?

Too much tape. 3 revolutions max, I use teflon tape every day at work, no need for that much. You obliterated the threads. Get a dremel and start cutting it out. You might try bolting the BB tool in place using the spindle, could get you a tighter/better fit.

Before you start cutting stuff, find a bolt that has the same thread as your BB spindle. Use it to loosely hole your BB remover flat against the BB splines. Then you can use a big wrench or maybe a cheater bar and it'll come right out.

OR the tape let it cross-thread. (More likely IMO.)

If a BB is torqued properly it won't keep loosening. Address the problem at its source.

The bolt you need is M8x1.0 about 40 or 50 mm long. This is a finer thread pitch than most common M8 bolts so you will need to find a well equipped hardware of home store.