Would Earth lose gravity during a pole shift?

When our Solar System passes through the Galactic Equator, yes it could play havoc with the laws of physics. Notice When in the Northern Hemisphere, and you flush the toilet, the water runs counter-clockwise. In the southern hemisphere it runs clockwise. This is due to the magnetic pole influence of the Earth.

I had typed in some snarky response originally but fortunately erased it.

Each sentence above is incorrect each in its own way.

"When our Solar System passes through the Galactic Equator, yes it could play havoc with the laws of physics."

*) The galaxy is not a solid rotating object unlike the Earth. There is not a perfectly well defined "equator". And the laws of physics work just the same given that we can observe stars close to some statistical consensus plane which you may define as the "equator".

"Notice When in the Northern Hemisphere, and you flush the toilet, the water runs counter-clockwise. In the southern hemisphere it runs clockwise. "

*) it does not. Water in the toilet or drain rotates in either direction due to whatever initial angular momentum it had.

*) Large scale weather patterns, namely cyclones, do have different rotation patterns in Northern and Southern hemispheres due to the Coriolis force, which is a consequence of observations on a rotating reference frame. This has significant effects at space scales larger than a few miles.

"This is due to the magnetic pole influence of the Earth."

*) There is no magnetic influence, it is a consequence of the relation to the axis of rotation of the Earth. If the magnetic poles 'flipped' North to South or started pointing at intermediate latitudes nothing would happen to either cyclones/hurricanes or toilets.