Agree that a low-cost, highly sturdy option would be a couple utility poles with pallets or planks of some sort on top. The problem with utility poles is they are HEAVY and you will not want to have to carry them far to place them. I have used 12" diameter utility poles for corner posts while fencing...an 8 foot piece weighs about 250-300 pounds...yours would be 600-700 pounds for a 16-18 foot pole. Not something you and your boys would want to carry 1/4 mile into a swamp/bog.
It would cost about $60 at Menards for a couple of 16 foot treated 2x8's...stand them on edge and screw pallets across them and you'd be set. The 2x8's would be MUCH lighter to carry and set and more than sturdy enough to walk across...I guess what I'm saying is cheaper isn't always better. $60-75 to build a simple foot bridge isn't expensive to me and will be MUCH easier to get in the woods than 16 foot utility poles in my opinion. If you aren't taking an ATV across them, I'd just use dimensional lumber and replace the pallets every couple years.
As for the neighbor...sounds like he may be trouble, but it could also be a front. I would maybe try talking to him again. Explain that since you are not from the area, you will be utilizing several cellular cameras to scout your new property and anyone seen trespassing will be prosecuted. There is his warning. You see them, call them in. If you can swing it financially, I would invest in a couple cell cameras if you don't have them. They are great scouting tools and are also eyes on your property when you are not around. Buy a section or two of hang on ladder sticks and hang them about 10-12 feet off the ground angled down...It will reduce the number of false triggers you get as well as deter theft as the cameras are above eye level and require a ladder or other tools to get down, which with a cell camera would give plenty of time for the camera to take and send a picture before they can steal it or turn it off. You can also reiterate that your kids will be hunting there and you don't want them injured or their hunts ruined. What I wouldn't do...is shrug it off and give them even an INCH of leeway once they are warned. Best of luck!