A threaded bottom bracket cup on the right, drive side of the bike does indeed unthread/loosen by rotating the cup clockwise. If this is a cartridge bearing bottom bracket, the body, bearing and spindle assembly--essentially the bulk of the bottom bracket--will come out the right side together with the cup. A loose ball bearing bottom bracket features the cups with their ball bearings separate from each other and the square taper spindle.
In any case you'll want to use the correct bottom bracket tool (almost always a splined tool whose number and shape of the splines mate perfectly into the receiving aspect of the cup. Especially on the drive side, it can take a great deal of force to break loose the cup's threads. While grease or anti-seize paste is best for thread treatment prior to installing a BB, some folks utilize a thread locker (aka Loctite) instead. Even a medium strength version of this stuff combined with achieving the appropriate torque of around 40 N-m make removing such a treated cup impossible without damaging something.
It's unclear whether you have got the drive side cup moving. If the cup is spinning in place without moving out, that is a bad sign. Could infer thread damage of the shell. Tapping the left side of the spindle or putting some lateral pressure there directed to the right while you rotate (clockwise) the right cup may give enough thread purchase to start the cup moving out.