In current affairs we see corporations threatening to close due to the new health care law regarding insurance coverage for medical procedures the corporate owners consider a violation of their religious views.
I know at the time of integration, there are some who cited religious reasons to continue segregation. I have seen this in transcripts of sermons.
But, did any businesses, similar to today, claim they would close their doors rather than abide integration for religious reasons?
Yes there were, but from what I can tell, it was mostly small businesses. The Sandoval-Strausz article (linked below) is an explanation of how racial integration was received by business owners opposed to integration.
Although public schools aren't a business, it's important to note that several schools and school districts shut down or became privatized as the result of racial integration, whether via the Brown v. Board decision or the Civil Rights Act of 1964. These schools were known as "segregation academies." For example, in Virginia, the Prince Edward County School system closed their schools for 5 years rather than integrate. This policy was part of a larger policy called Massive Resistance.
Sandoval-Strausz: http://www.jstor.org/discover/10.2307/30042844?uid=3739928&uid=2&uid=4&uid=3739256&sid=21103219136017
Massive Resistance in Virginia: http://www.vahistorical.org/collections-and-resources/virginia-history-explorer/civil-rights-movement-virginia/massive