No, the government should not provide tax benefits to companies who keep jobs in the U.S. Providing tax benefits and incentives to companies who threaten to outsource jobs will only set the example that all a company needs to do to get these rewards is threaten to ship jobs overseas.
American companies have been fleeing the US for decades; opting to move their office and production infrastructure to countries that offer the best tax-incentives for them to move operations there. So it only makes sense, and about time as well, that the US start competing on the level needed to not only retain the companies country here but bring back those companies that have left. If that means offering tax-breaks, then the US needs to offer tax-breaks.
No, the government should not provide tax benefits to companies who keep jobs in the US, but it should make it easier for companies to do so. The government should give tax incentives to the poor, not the wealthy. Companies must have a reason to stay, but it should not be through taxes.
Instead the government should punish companies that move jobs overseas by making them pay higher tariffs and taxes on goods and services that they important to the US. They should also ban the sale of many imported food by American owned companies that are trying to cut cost and higher less than expensive workers.