Software and service companies should work 24x7 like hospital

Actually this is true.  The expectation is high and competition is high.  Nowadays, especially SaaS based service companies, you are expecting the service is never stopped. Here is how we can accomplish:

• The company should work non-stop.  It does not mean everyone in the company has to be.
• Create multiple shifts.  People work in different hours and the total coverage will be 24 hours.
• Use remote resources.  People work from different time zones and thus by nature work in different hours.
• People work with remote team as a virtual team.
• Make the information transparent.  Post the progress and information to shared site.  Not just about the results, but about what change had made.
• Kanban, same idea as above.
• Transition during the day.  There are overlapping time zone or overlapping time between different shifts.  Make sure these are transferred:  What are the remaining work, what we have done, what you should avoid (we already tried, or we failed due to these)
• Be careful about rewarding systems.  Recognition of the efforts of working as a team
• Knowledge sharing and training.  If dividing work functionally, no one can take work from each other.  People need to be able to take similar jobs so a resource will not be a bottleneck.

It could be very discouraging if you create a culture to start blaming each others.