Can I use coaxial cable for a 600' network extension?
You can, but it's a VERY poor solution. You've crossed into my professional sphere of operations.

Also, if using coax, get extenders that are intended to work over coax - twisted pair and coax are quite different. Both types of converters are made.

Even if the mere thought terrifies the heck out of you, fiber is the sensible solution  to this problem. It should not be terrifying. I have done long connections (using various, often expensive, adapters and line extenders) on coax, twisted pair and fiber. All three technologies will do the distance with appropriate converters at the ends (speed won't be great on either of the copper solutions though - perhaps 30-40 Mbit in my experience at those distances. Less with crappier electronics.) Only one of them will work consistently in the real world where lighting happens - that one will also work at gigabit (1000Mbit) or faster speeds.

Been there, done that, have the burnt hardware to show for it. Fiber has permitted calm to prevail when thunderstorms visit.

In most cases, rather than getting a "fiber media converter" you will be better served and out less money for a gigabit (or 10 gigbit if you are really need-for-speedy) switch with SFP (SFP+ for 10 gig) slot or slots. Two low-cost switches, SFPs and 200 meters of pre-terminated fiber will probably cost less than a pair of copper line extenders - two factors - one a simple issue of scale - there's a limited market for copper line extenders, while there's a vast market for Ethernet switches, SFPs and fiber - the second is that there's some specialized electronic voodoo going on in a good line extender, while the fiber system is stock-simple. Getting paid for your voodoo in a small market makes for a high price tag.

As for wireless, how much chainsawing are you willing to do? You don't need a huge hole to get 5 GHz through at 600 feet - 5 feet wide and high will do. Technically, that big at the mid-point and less going to each end will do (google Fresnel zone if interested in the details.) Shooting through trees is iffy, but shooting through a hole in the trees is not. If TRYING to shoot through trees without clearing a hole, your odds improve at 900 MHz, but the equipment is less common and more expensive than 5 GHz stuff for point to point.

The cable should be bonded to the electrical ground at the service entrance of each building. This should help with, but does not consistently prevent, in my experience, damage from lighting-induced surges. If you get a direct strike, nothing really survives that, but that is not common. Fiber has the advantage of being completely electrically isolated, as well as being  much faster connection. It's the best network surge protection money can buy...

As for laying the cable (of whatever type) on the forest floor, good luck with that. Rodents with teeth... I'd strongly suggest looking for a possible 5 foot hole through the trees, and chainsaw half a dozen if needed to make it, rather than flaking cable on the ground. Try a bright light at one end and looking from the other end in the dark.