Should there be a minimum time in service for nomination as the Sec of a branch?

Since we are a civilian controlled military, I don't think the highest positions, which exist outside the military, should be required to have served, first. These are largely administrative positions. The Chief of Staff for each service provides the Secretaries and the President with the information they need to perform their duties.

No, not at all.. the position of secretary would be better served by an accomplished businessman and politician. Like it or not, those are the qualities that will allow them to be successful. They have SME's on staff to help them understand the path.
Service in a branch at less than the career level will not give them any useful insight for doing the tasks assigned as SECNAV, Army, AF, ect.

I don't think it should be a requirement. Each Branch has its Civilian Controller. If they are prior Military Service Members they may not be able to make the decisions they need to make.