I posted on your other, but yes, they can squeeze out of smaller holes than you think they can. 6x6 definitely. 3x3, if they work long enough at it, they could eventually.
The four foot fence is iffy - I wouldn't do one that low. Some Nigi's could easily get over that and some couldn't. I have one of each. One of my Nigi does can easily jump 5 feet up.
If there are cross pieces of wood that gives them something to stand on that could make it easier to get out also. Goats LOVE to stand with their front feet up on something; if it's possible they will and that will put weight on/against the fence. They WILL rub against any fence so it has to be really sturdy - whatever you use.
The more play things goats have in their fence area the less likely they are to try and get out. A bored goat is an escape waiting to happen unless the fencing is high and strong. That is also one of the reasons that it's imperative to have more than one goat (which you do). Two or more goats help keep each other company.
My Nigi's go through very small holes... I agree that 6x6 is easy for them. I've seen mine go through a smaller opening than that but the opening was in a wire fence so it 'gave' a little while they were crawling through it.
Picket type fencing is generally kind of dangerous for animals, because it is easier than you might think for them to get a chin or neck or ankle or armpit up over the fence and then slip and it wedges itself tightly in between the picket tops and the animal is injured or worse.
If you HAVE to run animals against picket fencing, it is safer to put a board along the top so there are no longer gaps between the picket 'tips'.
I have 4' woven wire no-climb fence with 2x4" spacing held up with T-posts and is tight to the ground, partially buried in places to keep it level, and they've never gotten out.
I have 4 foot high no climb woven wire 2x4 opening horse fence. My Nigerians don't get out of that. Make sure there are no low spots they can crawl under. They will definitely do that. So far no jumping over the fence.
My fence is Red Brand woven field fencing with 6 inch verticle wire and graduated horizontal wire with wooden posts set 10 feet apart. We have on strand of electric a foot off the ground on the inside, one strand about 6 inches above the top, and a strand 6 inches above the bottom on the outside. We have four wethered Nigerian Dwarfs (15 months, 11 months, 4 month old twins and have had no problems with escapes. There are no gaps under the fence anywhere at all. I think the electric really helps, although each goat has only gotten zapped a couple of times at the most. We have one of the gates with the graduated mesh on it, just like the field fencing except much heavier gauge. They have a quarter of an acre for pasture, but I just make sure there is nothing anywhere close to the fence that could be used to facilitate jumping over the fence. When I was building their new goat house, the top of the platform was 3 feet off the ground, only one could make it up on top of it while I was building. The other three would get close, but not be able to do it. Apparently mine aren't good jumpers, LOL. Hope this helps.