I have a civilian logbook and nothing for military time. SWA was happy with both. They read my civilian logbook in detail. I was surprised at that since it was all bugsmasher time. I had an excel spreadsheet printout showing how I converted my AF flight summary with .3 conversion. No questions asked there.
Personally, I regret not keeping a military logbook. My dad's logbooks are interesting reading, especially during Korea. I've denied that benefit for my kids.
You need a logbook regardless. Get a nice one and document all your upcoming flights (ATP, etc.) Do not let your IP's write in it. They can endorse, sign off, but leave out the commentary unless it is good stuff. Interviewers will read it. You're the customer so tell your instructor that before turning them loose with your log. "This is my logbook, not my gradebook..." is a good start.
-Elvis