Companies will go wherever they can pay the least amount of tax. Unfortunately that is the nature of the big corporations, they refuse to pay fair amount of taxes. What the EU are trying to do is brilliant, and for the sake of society let's hope they succeed. The challenge is that the companies will keep running, and it doesn't look like the U.S. government have the same appetite to make companies pay what they owe.
Companies that left the United States to domicile in low tax countries are likely to come back home. This was merely paperwork to change the headquarters. However, the European Union has levied hefty tax charges on companies such as Apple, and is investigating IKEA. Once the tax advantage is taken away, companies will have no need to change their headquarters outside the U.S.
Companies that are being taxed euros by the EU may choose to domicile in the U.S. Although America has a higher corporate tax rate, there are many ways companies can avoid U.S. taxes, so that the effective tax rate is much lower. But, companies may choose to domicile in other countries.
No, companies will not return to the United States simply because they are facing tougher tax scrutiny in Europe. The companies recognized that these taxes were due when they first located in Europe. Although the one-time tax bill for non-payment is large, the regular taxes are still less burdensome than the tax bills the companies face in the United States. Additionally, the tax burden is only one of many reasons that companies relocate. The cost of paying worker salaries in the United States, the insurance costs, and the high cost of lawsuits continue to discourage companies from domiciling in the United States.