"The gold and the silver are mine" - Haggai 2:8.
In the final analysis, nothing is Caesar's. Everything is God's. He holds the deed to the Universe. When Caesar tries to force us to his will instead of God's, God himself tells us to refuse.
"We must obey God rather than men" - Acts 5:29.
Perhaps you think this woman should leave her job because of her Christian convictions. Would you like every believing Christian to walk away from every government job that forces them to compromise their faith? I don't think you would like our nation with a government of only non-believers.
Better she should stand and fight. I support her 100%. She is standing up for God.