If you're using SOLID pipe (vs. perforated pipe) and your 'run' to the street (or wherever) is downhill, THen you don't need to go below frost line. In THIS scenario, you DON'T need gravel. You WILL need to dig down deep enough so that the TOP of the pipe is about 4-6" from the top of the ground. This is enough for grass to grow on. So, if you're using 5" pipe, then dig 10" down, give or take, and go from there.
Now, if your land is level or on a VERY slight downhill, then the TOP of the pipe needs to be at, or below, frost line. Secondly, you'll need a layer of Landscape Fabric in the ditch, followed by 4" of gravel, followed by the pipe (whose TOP is at frost line), then backfill all the way w/dirt (and plant grass.) Here's a link to get you started on this particular type of job. How To Construct A French Drain :(The actual link is down towards the bottom of the WEB page ...)
You'll also need to consider if you're using flexible pipe or SOLID pipe. If where you're digging is rocky, and I mean rocky in the sense that you may need to run your piping around buried rocks or tree roots or the like, then use flex pipe. If you're confident w/solid pipe, go for it. First, I'd dig the trench to see which type of pipe you'll need.
If you just do 'what you want to do' (which is dig just deep enough to bury the pipe), you may have problems. I'm just giving you heeded advice, that's all. You didn't say how deep the Pro was digging, you didn't say what type of pipe he's using, you didn't say what he is exactly 'tying' into, among other things. So, it's hard to tell if $600 is a 'fair price'. And to boot, pricing varies from local to local. Digging in Boston is MUCH more expensive than digging in SE PA.
That's what I think. My best to ya and hope this helps.
Jay J -Moderator
PS: Be SURE to use Landscape Fabric or even Filter Fabric if you're lining a trench w/gravel. Read the Link I gave you so you can see its purpose ...